A direct measurem ent of the W decay w idth

August 2008

Troy William Vine University College London

A thesis submitted to the University of London for the degreeof Doctor of Philosophy

I, Troy William Vine, confirm th a t the work presented in this thesis is my own. Where information has been derived froip. other sources, I confirm that this has been indicated in the thesis

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A bstract

A direct measurement of the W boson total decay width is presented in proton-

antiproton collisions at y/s = 1.96 TeV using data collected by the CDF II de­

tector. The measurement is made by fitting a simulated signal to the tail of

the transverse mass distribution in the electron and muon decay channels. An

integrated luminosity of 350 pb - 1 is used, collected between February 2002 and

August 2004. Combining the results from the separate decay channels gives the

decay width as 2.038 ± 0.072 GeV in agreement with the theoretical prediction

of 2.093 ± 0 .0 0 2 GeV.

A system is presented for the management of detector calibrations using a rela­

tional database schema. A description of the implementation and monitoring of

a procedure to provide general users with a simple interface to the complete set

of calibrations is also given.

A cknow ledgem ents

I wish to thank everyone who helped and supported me in the course of my PhD,

in particular my supervisor Mark Lancaster. Everyone else in the UCL group have

also been of great help in particular Dave Waters, Emily Nurse, Ilija Bizjak, Sarah

Malik and Dan Beecher. Also all those whom I worked with in calibrations have

always been eager to impart knowledge in an area tha t is largely undocumented,

in particular Ben Cooper and Pasha Murat. I wish to thank Nick Thomas for

inspiring me to do this PhD in the first place.

Funding from the Particle Physics and Astronomy Research Council and Fermi

National Accelerator Laboratory is gratefully acknowdedged with particular thanks

to Mike Lindgren.

3

C ontents

1 Introduction 19

1 .1 Standard m o d e l......................................................................................... 2 1

1 .1 .1 Standard model p a rtic le s ............................................................ 2 2

1 .1 .2 Quantum electrodynam ics......................................................... 23

1.1.3 Quantum chrom odynam ics......................................................... 24

1.1.4 Electroweak in teraction ................................................................ 25

1.1.5 Feynman d ia g ra m s ...................................................................... 29

1.1.6 C oupling.......................................................................................... 30

1.2 Drell-Yan process in hadron collisions................................................. 31

1.3 W boson production and d e c a y ........................................................... 35

1.3.1 W boson decay width prediction............................................... 38

1.3.2 W boson decay width m e asu re m e n ts ...................................... 40

4

2 E xperim ental apparatus 42

2.1 Particle interaction and d e te c t io n ......................................................... 43

2.1.1 Particle in te ra c tio n ...................................................................... 43

2.1.2 Particle d e te c tio n ......................................................................... 46

2.2 Fermi National Accelerator L a b o ra to ry ............................................... 47

2.2.1 T ev a tro n ......................................................................................... 49

2.3 Collider detector at Ferm ilab................................................................... 50

2.4 T rack in g ...................................................................................................... 51

2.4.1 Silicon t r a c k in g .............................................................................. 53

2.4.2 Central outer t r a c k e r .................................................................... 53

2.5 C a lo rim e try ................................................................................................ 56

2.5.1 Central electromagnetic calorim eter........................................ 57

2.5.2 Central hadronic calorim eter..................................................... 58

2.5.3 Plug ca lo rim ete rs .......................................................................... 59

2.6 Muon ch am b ers ......................................................................................... 60

2.6.1 Central muon d e te c to r ................................................................ 62

2.6.2 Central muon u p g r a d e ................................................................ 63

2.6.3 Central muon extension ............................................................. 63

2.7 Data acquisition......................................................................................... 64

5

2.7.1 Level 1 trig g e r ................................................................................ 64

2.7.2 Level 2 tr ig g e r ................................................................................ 6 6

2.7.3 Level 3 trig g e r ................................................................................ 6 6

2.7.4 Offline reconstruction .................................................................. 6 6

2.8 Lepton trigger paths and algorithms .................................................... 67

2.8.1 Extra fast tracker and extrapolation u n i t ............................... 67

2.8.2 Electron c lu s te r in g ...................................................................... 67

2.8.3 Electron trigger paths ................................................................ 6 8

2.8.4 Muon trigger p a t h s ....................................................................... 70

2.9 D a ta s e ts ...................................................................................................... 72

3 C alibration m anagem ent 73

3.1 Types of c a lib ra tio n s ............................................................................... 74

3.2 Database s t r u c tu r e .................................................................................. 75

3.2.1 Calibration retrieval ................................................................... 75

3.3 Offline procedure for ca lib ra tio n ............................................................. 78

3.4 Monitoring and validating c a l ib ra t io n s .............................................. 81

4 Event selection and reconstruction 82

4.1 Electron se lec tio n ..................................................................................... 83

6

4.2 Muon s e le c t io n .......................................................................................... 85

4.3 Recoil reco n stru c tio n ................................................................................ 89

4.4 Boson mass reconstruc tion ...................................................................... 94

4.4.1 W boson tut reco n stru c tio n ...................................................... 94

4.4.2 Z boson mass reconstruction...................................................... 95

4.5 Boson s e le c t io n .......................................................................................... 95

4.5.1 Z boson se le c tio n .......................................................................... 97

5 Event generation 98

5.1 Event g e n e ra t io n ........................................................................................... 100

5.1.1 Electroweak corrections and uncertainties...................................100

5.1.2 PDF u n c e r ta in ty ..............................................................................103

5.1.3 W boson mass u n c e rta in ty ............................................................. 105

5.2 Boson transverse m om entum .......................................................................105

5.2.1 Z boson transverse m om entum .......................................................106

5.2.2 W boson transverse m o m en tu m ................................................... 109

5.2.3 Boson transverse momentum u n c e rta in ty ...................................109

6 D etector sim ulation 113

6.1 z vertex s im u la tio n ....................................................................................... 114

7

6.2 Silicon tracker sim ulation............................................................................. 115

6 .2 .1 Silicon tracker material sc a le ..........................................................117

6.2.2 Silicon tracker simulation u n certa in ty ......................................... 118

6.3 Central outer tracker s im ulation .................................................................119

6.3.1 Central outer tracker scale and reso lu tion ...................................119

6.3.2 Central outer tracker simulation u n c e r ta in ty ............................ 1 2 1

6.4 Calorimeter s im u la tio n .................................................................................123

6.4.1 ToF, solenoid and leakage energy loss sim ulation .................... 124

6.4.2 Calorimeter response non-linearity .............................................124

6.4.3 Calorimeter response and reso lu tion .............................................126

6.4.4 Calorimeter response and resolution u n c e rta in ty ......................130

6.4.5 Electron E t sim ulation ....................................................................131

6.4.6 Electron E \ ^ / E em requirement s im u la tio n ............................... 132

6.4.7 Muon Eem requirement sim ulation...............................................132

6.5 Lepton selection simulation ....................................................................... 133

6.5.1 Lepton 77 and (p dependent selection sim ulation.........................133

6.5.2 Lepton u\\ dependent selection sim ulation...................................138

6.5.3 Electron E t dependent selection simulation ............................ 140

6.5.4 Muon pT dependent selection s im u la tio n ...................................143

8

6 .6 Recoil s im u la tio n .........................................................................................144

6.6.1 HEt s im u la tio n ................................................................................. 144

6 .6 .2 Soft recoil resolution s im u la tio n ....................................................146

6.6.3 Recoil resolution and response sim ulation...................................147

6.6.4 Recoil simulation for W boson e v e n ts ..........................................149

6.6.5 Recoil simulation u n c e r ta in ty .......................................................151

7 Background events sim ulation 156

7.1 Electroweak background e v e n t s .............................................................. 157

7.1.1 W t v b ack g ro u n d ........................................................................157

7.1.2 Z /7 * —> I t background e v e n t s ....................................................... 158

7.2 Decay-in-flight background e v e n ts ........................................................... 161

7.3 QCD background e v e n t s ............................................................................164

7.3.1 W —+ iiv QCD background e v e n ts ................................................ 165

7.3.2 W —> ez/ QCD background e v e n ts ................................................ 168

8 R esu lts 173

8.1 Fit r e g io n ...................................................................................................... 174

8.2 Fit r e s u l t s ...................................................................................................... 175

8.3 Related m easu rem en ts ............................................................................... 177

9

8.3.1 W boson lifetime . .

8.3.2 CKM matrix unitarity

Conclusion.................................

List o f Figures

1.1 W boson transverse mass distribution.................................................. 21

1.2 Feynman diagram of LO boson production and decay...................... 29

1.3 Feynman diagram of W boson decay with a radiated photon. . . . 29

1.4 Feynman diagram of boson decay with a virtual photon................. 30

1.5 Part on model of Drell-Yan W boson production............................... 31

1.6 The CTEQ6M PD Fs...................................................................................... 32

1.7 Feynman diagram of W boson production from gluons.................... 33

1 .8 Parton momentum fractions as a function of rapidity...................... 34

1.9 W boson production with initial state radiation................................ 36

1.10 Parton level differential cross-section.................................................... 37

1.11 Cross-section as a function of rax .......................................................... 38

1 .1 2 Summary of direct W boson decay width measurements................. 40

2.1 The Fermi National Accelerator Laboratory....................................... 48

11

2.2 Initial instantaneous luminosity of data used in this analysis. . . . 49

2.3 Elevation view of half of the CDF detector........................................... 51

2.4 Tracking volume of the CDF detector..................................................... 52

2.5 Central outer tracker cell layout............................................................... 54

2 .6 Central outer tracker superlayers............................................................. 55

2.7 Central calorimeter wedge.......................................................................... 57

2.8 Plug calorimeters.......................................................................................... 59

2.9 Muon subdetector coverage in the rj-cf) plane......................................... 61

2.10 Transverse view of a CMU wedge............................................................. 62

2.11 Trigger dataflow............................................................................................ 65

3.1 Calibration database schema..................................................................... 76

3.2 Pass evolution with calibrations used in this analysis......................... 79

4.1 The average tower energy deposit for leptons....................................... 91

4.2 Underlying event energy dependence on u\\ and luminosity............... 93

4.3 Underlying event energy dependence on 77............................................. 93

5.1 Non-resonant electroweak contributions to W boson production. . 101

5.2 Fractional change in m y from non-resonant contributions.....................102

5.3 Variation on Tw from the uncertainty on the PDF parameters. . . 104

12

5.4 Fitted p \ for boson pr simulation...............................................................108

5.5 ^ 7 for on mass-shell and highly virtual W bosons...............................109

5 .6 Confidence level contours for the g2~B covariance m atrix....................I l l

6.1 z vertex distribution.......................................................................................115

6.2 Fitted E /p for the SiliM ap material scale factor..................................117

6.3 Track curvature resolution............................................................................ 120

6.4 Fitted m ^ for pt scale and resolution......................................................121

6.5 Fitted E /p for the non-Gaussian A p scale factor...................................122

6 .6 Electron and photon energy loss in the ToF, solenoid and CHA. . 125

6.7 F itted E /p for calorimeter response non-linearity.................................. 127

6 .8 < E /p > for correlated calorimeter resolution........................................... 128

6.9 Fitted E /p for calorimeter response and uncorrelated resolution. . 129

6.10 Fitted for calorimeter response and uncorrelated resolution. . 130

6.11 Ehad/Eem distribution in Z —> ee events.................................................... 132

6.12 ELECTR0N_CENTRAL_18 trigger efficiency as a function of ?7trk* • • • 134

6.13 Electron and muon tack hit efficiency........................................................ 137

6.14 Lepton p and (f) distributions........................................................................138

6.15 Lepton selection efficiency as a function of.....u\\..................................... 139

13

6.16 Electron selection efficiency as a function of E T ......................................141

6.17 Muon (E/had) and selection efficiency as a function of Pt ...................... 143

6.18 EE't as a function of luminosity.................................................................. 145

6.19 Fitted E F T and (E £ t ) for simulation param eters..................................146

6.20 Fitted soft recoil resolution for simulation param eters.......................... 147

6.21 Recoil fits used to obtain the recoil parameters.......................................149

6.22 Recoil distributions for Z boson events..................................................... 150

6.23 Recoil distributions for W boson events....................................................153

6.24 Recoil distributions for W boson events....................................................154

6.25 Values of Tw obtained from the recoil covariance m atrix..................... 155

7.1 ttt-t distribution of W —► t v background events...................................... 158

7.2 F itted rriT distributions of Z/y* —► i t background events.....................160

7.3 Xtrack/nctf versus do for decay-in-ffight muons..........................................162

7.4 Fitted Xtrack/n<4f f°r decay-in-ffight background fraction...................... 163

7.5 Fitted rriT for decay-in-flight background events.....................................163

7.6 distribution of QCD background and W —> pv events.................... 166

7.7 W —> [11/ QCD background fractions.......................................................... 167

7.8 Fitted for W —> iiv QCD background events.................................... 168

14

7.9 distribution of QCD background and W —> eis events.....................170

7.10 Fitted rax for W —> ev multi-jet background events...............................171

8.1 rax distribution of background events.......................................................175

8.2 Fitted mx distribution to obtain Tw........................................................ 176

15

List o f Tables

1 .1 Standard model fermions....................................................................... 23

1.2 Summary of direct W boson decay width measurements.............. 41

4.1 Requirements for electron candidates................................................. 8 6

4.2 Track and calorimeter requirements for muon candidates............. 8 8

4.3 Stub requirements for muon candidates............................................ 89

4.4 Z veto requirements for additional track........................................... 96

4.5 W boson requirements........................................................................... 97

4.6 Z boson requirements............................................................................. 97

5.1 Systematic uncertainties from electron final state QED radiation. 101

5.2 Systematic uncertainties from muon final state QED radiation. . . 101

5.3 The correction to Tw from non-resonant electroweak corrections. . 103

5.4 Systematic uncertainties from the PDF param eters......................... 104

5.5 Systematic uncertainties from higher order PDF calculations. . . . 105

16

5.6 Systematic uncertainties from the W mass uncertainty........................105

5.7 Parameter values for the boson p r parameterisation............................ 108

5.8 Systematic uncertainties from the p™ simulation............................112

6.1 Systematic uncertainties from the energy loss simulation............. 118

6 .2 Systematic uncertainties from the silicon material scale................119

6.3 Systematic uncertainties from the p r scale and resolution............123

6.4 Systematic uncertainties from CEM response non-linearity..........126

6.5 Systematic uncertainties from the CEM response and resolution. . 131

6 .6 Parameter values for the XFT trigger efficiency.............................. 134

6.7 Parameter values for the electron track selection efficiency...........136

6 .8 Parameter values for muon track hit efficiency....................................... 137

6.9 Systematic uncertainties from lepton acceptance............................ 138

6.10 Parameter values for the e(u\\) simulation......................................... 140

6 .1 1 Systematic uncertainties from e(it||) simulation............................... 140

6 .1 2 Parameter values for e(Er) simulation..................................................... 141

6.13 Systematic uncertainties from e(£ r) simulation.....................................143

6.14 Parameter values for TjE t simulation....................................................... 145

6.15 Parameter values for the recoil response and resolution.......................148

17

6.16 Mean u\\ for data and simulation............................................................... 151

6.17 Systematic uncertainties from the recoil simulation.............................. 152

7.1 W t v background event fraction.......................................................... 157

7.2 Systematic uncertainties from W —> t v background events..................158

7.3 Z /7 * —*■ i t background event fraction....................................................... 159

7.4 Parameterisation of Z /7 * —> i t background raT distribution. . . . 160

7.5 Systematic uncertainties from to Z —> fifi background events. . . . 161

7.6 Systematic uncertainties from decay-in-flight background events. . 164

7.7 Systematic uncertainties from QCD background events...................168

7.8 Ratio of anti-electron to standard electron candidates......................169

7.9 QCD electron background event fraction.................................................171

7.10 Systematic uncertainties from QCD background events.......................172

8.1 Summary of all systematic uncertainties..................................................174

8 .2 Combined systematic and statistical uncertainties................................ 175

8.3 Number of background events.................................................................... 175

8.4 Summary of uncertainties, including correlations.................................. 177

18

Chapter 1

Introduction

The theory describing the interactions of fundamental particles is the standard

model. It is a quantum gauge field theory comprising the Glashow-Weinberg-

Salam model [1, 2, 3] of electroweak (EWK) interactions, unifying the weak and

electromagnetic interactions, and quantum chromodynamics (QCD) [4] describing

the strong interaction. The electromagnetic force is carried by a massless photon,

the weak force by massive W and Z particles, and the strong force by 8 massless

gluons. Direct evidence for the gluon was first observed by the TASSO experiment

using the PETRA accelerator at DESY in 1979 [5]. This was followed by the

discovery of the W [6 , 7] and Z [8 , 9] bosons by the UA1 and UA2 experiments

at CERN in 1983. This provided strong evidence in support of the standard

model. The only subsequent experimental evidence indicating tha t the standard

model is not complete is the observation of neutrino oscillation [10]. The unified

electroweak theory predicts the existence of the Higgs field [11] to break the

symmetry between the electromagnetic and weak interaction resulting in massive

bosons for the weak interaction. The associated Higgs boson has not been directly

19

observed and is the next crucial test for the standard model. Since the standard

model is only consistent with special relativity and not general relativity, it is not

a complete theory of fundamental particles and does not include gravitational

interactions.

Since the number of possible experimental measurements is greater than the

number of free parameters, the standard model is an ‘over-constrained’ theory

and precise experimental measurements provide stringent tests of the consistency

of standard model predictions. One example, and the subject of this thesis, is the

W boson decay width which is predicted at tree-level by the W boson mass and the

Fermi constant, described in section 1.3.1. A high precision direct measurement

of these quantities provides a crucial test of EWK predictions. Currently, the

largest uncertainty on this constraint is due to the uncertainty on the W boson

decay width measurement.

The distribution of virtual W bosons is sensitive to new physics. For example,

the distribution of the W boson transverse mass, defined in equation 1.21, has

been measured up to 950 GeV [12] at the Collider Detector at Fermilab (CDF)

experiment, described in chapter 2. The agreement with the standard model

prediction, shown in figure 1.1 together with the signal of a 800 GeV hypothetical

W' boson, excludes the existence of an additional boson with a mass below 800

GeV and the same decay channels as the W boson [13]. Additional decay channels

for the W boson, such as an additional neutrino flavour, are not constrained by

this test.

This thesis presents a precision measurement of the W boson decay width at the

CDF experiment. This is compared with the standard model prediction of the

W boson decay width, described in section 1.3.1, using the latest measurement

20

Figure 1.1: The W boson transverse mass distribution (DATA) measured at CDF in W —> eis decays, together with the prediction for an 800 GeV W ' boson (signal prediction) and the standard model background (background prediction) [13].

of the W boson mass and the measured Fermi constant. Since the prediction of

the decay width is insensitive to new physics, see section 1.3.1, this measurement

provides a test of the consistency of the standard model.

This chapter describes the relevant theory for W and Z boson production in

hadron collisions, and their subsequent decay.

1.1 Standard model

The standard model describes the interaction of structureless fermions, which

have half-integer intrinsic angular momentum or ‘spin’, and their interaction via

integer spin bosons.

21

1.1.1 Standard m odel particles

The propagation of a fermion field is described by the free Dirac equation, with

Lagrangian

Dirac = ^ 7 ^ “ (1-1)

where ip are the Dirac spinors, which are four component column vectors, and m

is the mass of the associated field quantum.

There are four independent solutions to the above equation, two with positive

energy and two with negative energy. These are interpreted as particle and an­

tiparticle components respectively, which are identical except the sign of the

quantum numbers is reversed. Unless explicitly stated, antiparticles are implied

in the following discussion. The Dirac spinors can also be decomposed into ‘right

handed’ and ‘left handed’ components, which have positive and negative helicity

respectively, where the helicity is the component of spin along the direction of

motion.

The fermions undergo electromagnetic, weak and strong interactions due to an

electric charge, a weak isospin charge and a colour charge respectively. This is

described further below.

Only weak interactions can change the fermion type or ‘flavour’, and flavour

changing interactions only occur between left-handed fermions. The fermions are

therefore grouped into left-handed isospin doublets, with components tha t inter­

change under flavour changing weak interactions, and right-handed singlets. The

lepton doublet consists of an electrically charged lepton with charge q = — 1 , and

a neutral neutrino. The lepton has only one singlet, the right-handed charged

lepton, since the right-handed neutrino cannot be observed (in the massless neu­

22

trino hypothesis). The electric and weak isospin charge are combined to create

the weak hypercharge, described by the electroweak theory of the unified weak

and electromagnetic interactions.

Three ‘generations’ of fermions exist, shown in table 1.1, with the leptons com­

prising the electron (e) with a mass of 0.511 MeV, the muon (fi) with a mass of 106

MeV and the tau lepton (r) with mass 1.7 GeV and their associated neutrinos.

The three generations of quarks comprise up-type and down-type quarks with

electric charge + 2 /3 and —1/3 respectively. Quarks additionally carry a colour

charge, enabling QCD interactions described in section 1.1.3, and only the most

massive quark, the top quark, is observed in an unbound state with mass 172.4 +

1.4 GeV [14]. The other quark masses lie between the 1 MeV and 4.5 GeV.

Table 1 .1 : Standard model fermions grouped into left-handed weak isospin doublets and right-handed weak isospin singlets.

1.1.2 Quantum electrodynam ics

The quantum field theory of electromagnetic interactions is quantum electrody-

manics (QED) which describes the interaction of electrically charged fermions

via the exchange of a photon [15]. As energy is not conserved at the photon

vertex, the photon (7 *) is ‘virtual’ and reabsorbed at a second vertex to conserve

Leptons:

Quarks:

23

energy overall. The momentum transfer is limited by the Heisenberg uncertainty

principle.

As the phase of the fermion field is not observable, the field is required to be

invariant under a local phase transformation, described by the U (l) symmetry

group. This introduces a vector gauge field A^, whose associated quantum is the

photon. The QED Lagrangian is

C q e d = W Y D n - m ) ip - ( 1.2 )

where the covariant derivative (D^) is given by

D^ = d p - ieQ A ^ (1.3)

and eQ is the electric charge that couples to the photon field. The last term in

the Lagrangian, the photon field tensor

= dvA v - d^Ap (1.4)

gives the photon field kinetic energy. A photon mass term is not allowed as it

would break the gauge invariance.

1.1.3 Quantum chrom odynam ics

QCD, the quantum field theory of colour charge interactions [4], describes the

interaction of quarks and gluons, which carry a colour charge. The colour charge

exists in three possible ‘colours’, red, green and blue (antired, antigreen and

antiblue for antiquarks). By requiring the quark field to be invariant under local

24

colour transformations, described by the SU(3) symmetry group, eight gluon

vector fields G“ are introduced, where a = 1,..., 8 . The eight gluons carry a

colour charge which consists of a ‘non-colourless’ combination of a colour and

an anticolour (the colour singlet R R + GG + B B is not allowed). The QCD

Lagrangian is

Cqcd = Q i i^ D , - m)q - (1.5)

where the covariant derivative (D^) is given by

= d^ + igs TaG“ (1 .6 )

and gs is the gluon coupling strength to the colour charge and Ta are the eight

matrix generators of the SU(3) group of colour transformations. Again local gauge

invariance requires the gluons to be massless. The last term in the Lagrangian

gives the gluon field kinetic energy and the gluon field tensor G®u is

= d ,G l - dvG l - g J ^ G l G l (1.7)

where f a\>c are the SU(3) group structure constants. The last term in the gluon

field tensor is the result of the non-Abelian nature of SU(3) and represents gluon

self-interactions.

1.1.4 Electroweak interaction

Weak interactions [15] consist of ‘flavour changing’ charged current (CC) inter­

actions and neutral current (NC) interactions. The charged current interactions,

mediated via the W boson, couple to the weak isospin doublet which has a weak

25

/

isospin charge of T = 1/2. The neutral current interactions, mediated via the Z

boson, couple to both the weak isospin doublet and singlet. Since the singlet has

T = 0, the neutral current is split into two components, one that couples to weak

isospin and the other that couples to weak hypercharge Y = 2 (Q — T3), where Q

is the electric charge and T3 is the third component of weak isospin. Hence the

neutral current is a combination of weak and electromagnetic interactions.

The weak eigenstates (d', s', b') of the quarks are linear combinations of the mass

eigenstates (d, s, b) related by the Cabibbo-Kobayashi-Maskawa (CKM) matrix

below.

' Kd Vw Kb

b d b * b b s ( 1 . 8 )

b d b s b b

Experimental measurements of the matrix elements give diagonal elements close

to unity and small off diagonal elements, resulting in ‘Cabibbo’ suppressed inter­

actions between different quark generations. The CKM matrix is a unitary matrix

in the standard model, with the constraints JT = $jk and Ylj Vij^kj = $ik

where S^j)k is the Kronecker delta.

Weak and electromagnetic interactions can be described together by the S U ( 2 ) L 0

U ( l ) y symmetry group, where S U ( 2 ) L is the symmetry of left-handed weak

isospin, and U ( l ) y is the symmetry of weak hypercharge. The requirement of

gauge invariance gives three vector gauge fields WJ, where i = 1, 2, 3, and a sin­

gle vector gauge field B jl. The VC/ fields couple to weak isospin with strength g ,

and the field couples to weak hypercharge with strength g'. The covariant

vb'/ V vb/

26

derivative for left handed fermions is (for right-handed fermions g = 0 as T = 0)

D = dl> + ig T iW ; + ig’ '^ B li (1.9)

where T* are the matrix generators of the SU(2)l group of weak isospin trans­

formations. The mass eigenstate fields, associated with the observed massive W

and Z bosons, are mixtures of the and B M fields. The charged current fields

are

(1.10)

and the neutral current fields are

* \

\ A , j

—sinOw cos 6*w

V cos #w sin #w( l . i i )

where the Weinberg angle #w ~ 28° is found experimentally. The weak and

electromagnetic coupling constants are related by

e = g sin #w — g' cos 9w ( 1.12)

since A M is the electromagnetic field.

The interaction terms of the Lagrangian for the charged current W+ field (the

interaction term for the W ~ field is the Hermitian conjugate) are

£ c c = 92y/2

~lh)qd + ^ 7m(1 - 7s )/ ) (1.13)

27

and the interaction terms of the Lagrangian for the neutral current are

CNC = eA^ 'ipYQ'^ + 7;— ^ ~ z n 9v ~ 9Alb)^2 cos Bw

(1.14)

where the vector coupling gy — X3 — 2 Q sin2 6w and the axial-vector coupling

qa = T3 . As a result, neutral current interactions, such a fermion annihilation,

have a Z boson and photon component. However, when the centre-of-mass energy

is similar to the Z boson mass, a resonant condition occurs and the Z boson

exchange dominates.

The weak vector bosons have observed masses, but mass terms of the form

| m 2W ^W cannot be added to the Lagrangian, as it will break the Lorentz

invariance. Instead, terms that preserve the SU(2 ) gauge symmetry are added.

The ‘spontaneous symmetry breaking’ Higgs mechanism introduces a complex

doublet of scalar fields (f) that couple to the WJ and BM fields. This adds a po­

tential term V(<j>) = — A (0 ^ )2. By choosing /i2, A < 0 there is a non-zero

minimum at |0| = /i/y 2 |A |. Expanding about the minimum gives mass terms to

the electroweak fields with

and introduces the massive scalar Higgs boson. The mass of the Higgs boson is

not predicted a priori by the theory, although its mass can be constrained from

EWK measurements, particularly the mass of the W boson and top quark. The

observation and direct mass measurement of the Higgs boson are the subject of

continuing research.

(1.15)

28

1.1.5 Feynman diagrams

A scattering process can be expressed as a perturbative expansion of the Hamil­

tonian [16]. The terms in the expansion are multiplied by a power of the coupling

constant cm, referred to as the order. Each term may be represented by a Feyn­

man diagram. In the Feynman diagram, incoming and outgoing lines represent

fermions before and after scattering respectively. Internal lines and loops rep­

resent intermediate bosons, and boson radiation is represented by a boson line.

Each vertex introduce a factor of a/cm in the m atrix element, and hence a factor of

cm in the cross-section. Leading-order diagrams, such as W and Z boson produc­

tion and decay shown in figure 1 .2 , contain two vertices and represent the first

term of the perturbative expansion.

Figure 1.2: Feynman diagram of leading-order boson production and decay.

The additional perturbative terms provide corrections to the leading-order term,

and become increasingly suppressed with higher-order.

Figure 1.3: Feynman diagram of W boson decay with a radiated photon.

29

Radiative corrections, such as photon radiation from the boson or final state

electron in W boson decay shown in figure 1.3, contribute additional particles to

the final state. Virtual corrections, such as those shown in figure 1.4 for W and

Z boson decay, do not contribute any additional final state particles.

Figure 1.4: Feynman diagram of boson decay with a virtual photon.

Terms with an additional factor of a compared to leading-order terms, such as

single photon radiation, are referred to as ‘next-to-leading-order’ (NLO) correc­

tions. Similarly ‘next-to-next-to-leading-order’ (NNLO) corrections, such as two

photon radiation, contain an additional factor of a compared to NLO.

1.1.6 Coupling

The measured coupling of a charge to its associated field is the sum of the per­

turbation series. As the sum of the series is divergent, the sum must have a

scale-dependent cut-off. The ‘renormalisation’ of the Hamiltonian provides a

measurable, scale-dependent coupling [16]. The coupling of QED interactions

increases with the square of the momentum transfer (Q2), while the coupling of

QCD interactions (a s) decreases with Q2 due to gluon self-interactions. Quarks

are therefore ‘asymptotically free’ at high energies and can be treated as free

particles. At larger distances, a s increases and quarks are confined into ‘colour­

less’ combinations of quarks, known as hadrons. Bound quarks may be quark-

30

antiquark pairs, which have a colour and anti-colour charge respectively (eg red

and anti-red), or three quark combinations with have equal amounts of red, green

and blue, such as a proton comprising the uud quark combination.

1.2 Drell-Yan process in hadron collisions

W and Z bosons are produced at hadron colliders by the Drell-Yan process where,

at tree-level, individual constituents (partons) within the colliding hadrons an­

nihilate to produce the boson. This parton model of ‘hard’ collisions is shown

in figure 1.5 for the leading-order Drell-Yan W boson production at a proton-

antiproton (pp) collider. The other ‘spectator’ partons, shown as double lines,

form final state hadrons.

UUd

uud

Figure 1.5: Parton model of Drell-Yan W boson production.

For a colliding proton and anti-proton with 4-momentum Pp and Pp respectively,

the Mandelstam variable s = (Pp + Pp) 2 is Lorentz invariant and equal to the

square of the centre-of-mass energy. Neglecting the proton rest mass, s « 2Pp Pp.

31

The centre-of-mass energy of the two interacting partons (s ) is related to the

pp centre-of-mass energy by s = X1X2 s, where x\$ are the momentum fractions

carried by the two interacting partons. For leading-order boson production, the

boson 4-momentum q is given by q2 = s.

The structure of the proton is described by the parton distribution functions ( /q)

which satisfyall partons „ j

y ; / xfq{x, Q2)dx = 1 (1-16)Jo

where Q is the scale. The parton distribution functions (PDFs) cannot be de­

termined theoretically as the partons are in a bound state, and are measured

experimentally using global fits to hadronic data [17]. The CTEQ6M PDFs corre­

sponding to a scale Q = raw are shown in figure 1 .6 .

102

10

1

up down strange charm

X

Figure 1.6: The proton x f q distributions at Q = raw for the CTEQ6M PDFs.

Since the Tevatron, described in section 2 .2 .1 , has a centre-of-mass energy y/s =

1.96 TeV, a W boson produced at rest with q2 = ra ^ results from both partons

carrying momentum fraction x « 0.04. At this value of x, a considerable amount

of the momentum is carried by the ‘valence’ quarks which make up the quantum

numbers of the proton. At a hadron collider such as the Large Hadron Collider

32

(LHC), with a centre-of-mass energy y/s = 14 TeV, the associated momentum

fraction is x ~ 5 x 10~3 and the gluon is the most prevalent parton at this value

of x.

In addition to ud interactions creating W + bosons (henceforth antiparticle equiv­

alents will be implied), cs, us and cd interactions also occur, although with less

frequency. This is because the charm and strange quarks have a lower probability

density in the proton structure and us and cd interactions are suppressed due to

the small off diagonal elements of the CKM matrix. Gluon interactions, such as

dg and gg shown in figure 1.7, also occur in W boson production although these

interactions are suppressed due to the coupling of the additional gluon vertex.

(I

9 a u

Figure 1.7: Feynman diagram of W boson production from gluons.

As the momentum fractions of the two partons are not necessarily equal, the

boson can have a non-zero momentum component (pz) in the proton direction.

The rapidity (Y ) of the boson is given by

r = 5 ln W

where E is the boson energy. The rapidity between two particles is invariant

under a Lorentz transformation parallel to the proton direction. The parton

33

momentum fractions are related to the boson rapidity by

XU = y j \ ^ Y ( 1. 18)

shown in figure 1.8 for Vs = raw at the Tevatron. Events selected with a ‘min­

imum bias’ are produced uniformly in rapidity and are the most frequently oc­

curring interactions at hadron colliders.

X 0.80.7

0.6

0.5

0.4

0.3

0.2

0.1

■3 2 ■1 0 1 2 3

Figure 1 .8 : Parton momentum fractions as a function of rapidity.

For a massless particle, the rapidity is equal to the pseudorapidity (77) given by

77 = — In ( tan ^ ) (1-19)

where 9 is the angle between the particle and the proton direction.

The invariant mass of two particles produced by the decay of a massive particle

is equal to the rest mass of the latter, given by m = yj(P a + Pb)2 where Pa^ is

the 4-momentum of the decay products. This can be expressed as

m = V + £b)2 - (pa + Pb)2 (1-20)

34

where and pa,b are the energy and momentum vector respectively of the final

state particles a and b.

Although the Z boson’s invariant mass is reconstructed at hadron colliders, this is

not possible for W bosons as the neutrino, and hadron remnants at large rapidity,

are not detected. Instead the transverse mass of the W boson is reconstructed.

W bosons produced at rest in the detector decay with the charged lepton pT equal

and opposite to the neutrino pt-

The boson mass in the transverse plane (mT) is calculated from the transverse

momentum P t = P sin 9 of the decay products. Prom equation 1.20, the transverse

mass of two particles with transverse momentum pj, and pj, and azimuthal angle

A ( f ) between them is given by

neglecting the rest mass of the final state particles.

Higher-order Drell-Yan production processes result in ‘initial-state’ radiation from

the interacting partons or the W boson, giving the latter a transverse momentum

component. Photon, quark and gluon radiation from the colliding partons is

shown in figure 1.7 and 1.9. Photon radiation from the W boson is shown in

figure 1.3 (right).

( 1.21 )

1.3 W boson production and decay

The Drell-Yan cross-section can be expressed as a ‘hard’ parton-level subprocess

convoluted with the PDFs f q and f q of the proton and anti-proton respectively.

35

Figure 1.9: W boson production with initial state (left) photon and (right) gluon radiation.

At leading-order, this can be expressed as

all partons „ \a = y z / ^ M xi^Q2) M x2^Q2)dx1dx2 (1.22)

Jo

where a is the cross-section of the parton-level subprocess describing the produc­

tion and subsequent decay of the boson.

To describe the parton level cross-section for a boson of mass ra, the relativistic

propagator q2 — m 2 must be modified to describe the boson decay. The finite

boson lifetime results in an uncertainty in the boson mass (Tw)- This is related

to the proper lifetime r by

r = - (i.23)T

where h is the reduced Plank constant.

While it is sufficient to use the propagator q2 — m 2 + zraT near the pole at

q2 = raw, the effects of higher-order corrections to highly virtual bosons are

properly included by using q2 — ra2 + iq2T /m [18]. The denominator of <r contains

the square of the propagator factor and, using q2 = s, the s-dependent Breit-

36

Wigner cross-section can be expressed as

m w ( s - n i v v ) 2 + (srw/mw): (1.24)

where r qq and T/ are the partial decay widths into the initial state and final

state respectively. The cross-section as a function of y/§ is shown in figure 1.10

for ud —> W + —> e+ + ve.

10'5

100 150 200Vs (GeV)

Figure 1.10: The s differential parton level cross-section for ud —-»• W + —► e + ve.

Real W bosons are produced near the pole at y/§ = raw, and have a cross-

section that is independent of Tw, as r qq r f oc Conversely, virtual W bosons

produced at Vs mw have a cross-section that is approximately proportional

to Hence a comparison of a for real and virtual W boson production gives

a measurement of Tw . It is not possible to measure <r or s directly at hadron

colliders, rendering this approach impossible, but the related variables a and m T

are well measured, and a as a function of % , shown in figure 1.11 simulated

for different values of Tw , is sensitive to the value of rw. The cross-section is

affected by higher-order corrections, PDFs, detector acceptance and resolution,

and event contamination from other processes. Hence Tw cannot be extracted

37

from the data analytically, and a Monte Carlo simulation of the cross-section as

a function of mT is used instead. The value of Tw is obtained by comparing the

simulated cross-section as a function of raT with data.

12.5 GeV

2.1 GeV1.6 GeV

50 100 150 200n^. (GeV)

Figure 1.11: The simulated W boson production cross-section as a function o f m T for different Tw values.

1.3.1 W boson decay w idth prediction

In the standard model, the W boson decay width is theoretically predicted from

the measured W boson mass and the measured Fermi constant. Any corrections

to the decay width due to new physics can be absorbed into the renormalisa­

tion of the W boson mass and the Fermi constant. As a result their impact on

the measurement of Tw is less than 1 MeV [19]. The W boson decay width is

evaluated below using the latest world average of the W boson mass.

At leading-order, the standard model partial decay width for W + —> e+ue is given

by0 _ g2m w _ GFm ^W 48tt 6ttC2 ( }

where GF is the Fermi constant. Using Gp = 1.16637 x 10-5 GeV-2 , measured

38

in muon decay [20], and the latest world average mw = 80.399 ± 0.025 GeV [21],

the partial decay width is given by

r (W + -> e+i/e) = (1 + <5ewk) r ^ - 226.47 ± 0.25 MeV (1.26)

including higher order electroweak corrections of <5ewk = —0.4% [19]. This cor­

rection is very small as most of the standard model electroweak corrections are

included in mw and G?.

The W boson can decay via three leptonic channels and two quark doublet chan­

nels. W boson decay to the top quark, for s ~ mw, has a negligible cross-section

due to the large mass of the top quark. The hadronic decay channels q{qj for W +

decay, where the quark mass eigenstates qi = u,c and q] = d, s, b, have partial

widths

r (W + -> qiq-) = 3 |v y 2(l + <5qcd + <5ewk) i y = (707.11 ± 0.92)\Vt3\2 (1.27)

where the factor 3 is the number of possible colour combinations and are the

CKM matrix elements. The QCD correction (£QCD) to third order in a s is

2 3S QCD = + 1 4 0 9 _ 12 77 = 0.0410 ± 0.0007 (1.28)

7T 7T TT6

using a s(Mw) = 0.120 ± 0.002 evaluated from the world average of a s(Mz) =

0.1176±0.002, and evaluated at Q = mw using the NNLO renormalisation group

equation [20].

Adding the two quark doublet and the three leptonic decay channels gives the

39

prediction of

Tw = 2.093 ± 0.002 GeV (1.29)

with an uncertainty dominated by the uncertainty on mw- This is in agreement

with the prediction of TV = 2.0910 ±0.0015 GeV using a value of mw determined

indirectly from EWK data [20].

1.3.2 W boson decay w idth m easurem ents

Direct measurements of the W boson width have been made at the Super Proton

Synchrotron (SPS), the Large Electron Positron Collider (LEP) and the Tevatron

accelerators, and are shown in figure 1.12 and presented in table 1.3.2.

LEP2

D0

CDF

[ | world average

1.8 2 2.2 2.4r w (GeV)

Figure 1.12: Summary of direct W boson decay width measurements with the shaded region showing the world average with the measurement presented in this thesis exlcuded (UA1 is not shown).

In addition to the direct method to measure the W boson decay width described

above, indirect measurements can be made from the cross-section ratio of

Experiment Year Value/GeV Uncertainty/GeVUA1 [2 2 ] 1989 2 .8 2 .0

CDF [23] 2 0 0 0 2.04 0.14D 0 [24] 2 0 0 2 2.23 0.18LEP2 [25] 2007 2 .2 0 0.08World average [25] 2007 2.15 0.06

Table 1.2: Summary of direct W boson decay width measurements, with the mea­surement presented in this thesis excluded from the world average.

where B r ( W In) = T( W -> ln) /T(W) and B r ( Z -> l+l~) - F(Z

l+l~)/T(Z). Using the value of R measured at CDF, together with the theo­

retical prediction of the total cross-section ratio and partial width T (W —> In)

and the measured value of B r ( Z —► the W boson decay width has been

indirectly measured as Tw = 2.08 ± 0.04 GeV [26]. While this provides the most

precise measurement of Tw to date, it is not strictly a measurement of the to­

tal W boson decay width. Hence does not offer independent verification of the

standard model.

41

Chapter 2

Experim ental apparatus

The data used in this analysis has been collected at the Collider Detector at Fermi

National Accelerator Laboratory (CDF) near Batavia, Illinois. CDF is located

on the Tevatron, a superconducting proton-antiproton accelerator. This exper­

iment has made many important discoveries and measurements in high energy

physics, including the discovery of the top quark using data collected between

1992-1996 [14]. The Tevatron has since undergone a major upgrade to increase

the centre-of-mass energy from 1.8 TeV to 1.96 TeV and to achieve a thirty fold

increase in luminosity. The CDF detector has also undergone a major upgrade

[27] and has continued to make important measurements and discoveries including

the observation of time-dependent Bg-Bg oscillations in data collected between

2002-2006 [28]. This chapter contains an overview of the CDF detector and a

detailed description of the subsystems used in this analysis.

42

2.1 Particle interaction and detection

2.1.1 Particle interaction

High energy particles are detected and tracked via their interaction with a medium.

Energy transferred via interactions with the medium leave it in an excited or

ionised atomic state, allowing detection. Energy loss mechanisms relevant to the

simulation, described in section 6 .2 , and particle detection techniques relevant to

the CDF analysis are briefly described below and can be found in more detail

elsewhere [29].

Ionisation

Virtual photon exchange between a fast charged particle and an atom in the

medium may result in the ionisation of the atom. The energy loss of the charged

particle as it travels through the medium is parameterised by the Bethe-Bloch

equation

dE__K_dx (32

1 2mec2(32~f2Tmax 2 S'2 P ---------0 ~ 2

(2 . 1)

where m e and c are the electron rest mass and speed of light in vacuo respectively,

(3 = v /c where v is the particle speed and 7 = 1/ y / l — (32. The constant factor

K , the ionisation potential I and the density effect correction 6 are dependent

on the material traversed. Ionisation is the dominant energy loss mechanism for

high energy muons, and has a maximum kinetic energy transfer of

43

_ 2mec2 /5 27 2

max = i _j_ 27 "^ + (nk.)2 (2'2)

where mM is the muon rest mass.

Brem sstrahlung

A charged particle may emit a high energy (hard) photon to conserve momentum

when scattered from the strong localised electric field of an atom. Bremsstrahlung

is the dominant energy loss mechanism for relativistic electrons. The probability

for radiation in a layer of material with fractional radiation length dX 0 is

P7 - d X o x - f i^ (? /m a x /2 /m in ) ( l /m a x 2 /m in ) T 0 ( U m a x P m i n ) (2.3)

where ym-m and ymax are the minimum and maximum fraction of the electron’s

energy transferred to the radiated photon.

Pair production

At high energy, photons in a medium can be converted into an electron-positron

pair providing the photon has energy P 7 > 2 me + P N where P N is the kinetic

energy gain of the recoiling atomic nucleus due to the transfer of the photon

momentum. The photon conversion probability is

P j - > e + e ~ = g (2-4)

44

in the high energy limit [2 0 ].

Electrom agnetic showers

Electrons traversing a medium with energy greater than about 10 MeV predom­

inantly interact via bremsstrahlung radiation. As a radiated photon may then

convert to an e+e- pair, a cascade of electrons, positrons and photons ensues.

This electromagnetic shower will develop until the radiated photons are below the

electron pair production threshold. Electromagnetic showers in the calorimeters,

described in section 2 .1 .2 , enable a measurement of the electron energy.

Hadronic showers

Hadrons interact with the atomic nucleus via the strong interaction. Hence the

resultant shower contains hadrons in addition to electrons, positrons and pho­

tons. The composition of hadronic showers varies considerably due to the many

different interactions that can occur. As the rate of energy deposition is parti­

cle dependent, the fluctuation in the energy measurement is greater for hadronic

showers than for electromagnetic showers of the same energy. Hadronic show­

ers are used to determine the energy of hadrons in the hadronic calorimeters

described in section 2.5.2. They are used in this analysis, together with electro­

magnetic showers, to measure the event recoil, described in section 4.3 and the

missing energy described in section 4.4.1.

45

2.1.2 Particle detection

Gas ionisation detectors

If the medium is normally a poor electrical conductor and the charge is able to

move freely, the ionisation due to the charged particle can be detected. Gases such

as Argon fall into this category. The gas is placed in a high electric potential and

the ions will drift towards the cathode where they are detected. Gas ionisation

detectors are best suited for tracking charged particles over large distances as

the rate of energy loss of the fast particle is small due to the low density of the

medium. The gas ionisation detectors at CDF are central to this analysis. They

are used to measure the momentum of electrons and muons in the central outer

tracker described in section 2.4.2 and to identify muons in the muon detectors

described in section 2 .6 .

Solid state ionisation detectors

Carefully chosen impurities added to silicon can act as an electron donor (n-

type silicon) providing an excess of electrons in the conduction band, or as an

electron acceptor (p-type silicon) creating an excess of positively charged mobile

‘holes’. A junction between these two silicon types will be traversed by electrons

flowing into the p-type region and by ‘holes’ flowing in the opposite direction

until the resultant potential across the junction blocks further charge migration.

The region at the junction will then be depleted in charge carriers and any charge

released in this area by ionising particles will migrate away from the depletion

area giving a detectable current across the junction. The size of the depletion

layer is increased by applying an external ‘reverse-bias’ electric potential across

46

the junction. Solid state detectors at CDF are the silicon detectors, described in

section 2.4.1 and used in this analysis to determine the beam position.

Scintillation detectors

Molecules in an excited molecular state due to a virtual photon exchange may

decay to the ground state via radiative de-excitation, where the energy is carried

off by an emitted photon. Scintillating materials are transparent to the emitted

scintillation light, allowing subsequent detection. Scintillating detectors are used

to detect charged particles in the calorimeter, described in section 2.5, and muons

in the muon chambers described in section 2 .6 .

Sam pling calorim eters

To determine the total energy of the particle it must be brought to rest in the

detector medium. As the rate of energy loss increases with atomic number, a

compact detector medium is obtained by alternating a scintillator with an ab­

sorber. The shower evolution is then determined from the sampling scintillator

layers by comparing them to test beam data and simulation. The calorimeters at

CDF are sampling calorimeters.

2.2 Fermi National Accelerator Laboratory

The Fermi National Accelerator Laboratory (FNAL) has many experiments using

the proton source at the heart of the laboratory, shown in figure 2.1. Protons and

antiprotons are provided for the Tevatron, described in section 2.2.1, where they

47

axe accelerated before being collided in the CDF and D 0 detectors. The protons

are also used to produce neutrinos for the Main Injector Neutrino Oscillation

Search (MINOS) and the Booster Neutrino Experiment (BooNE), and fixed target

experiments. The production and acceleration of protons and antiprotons for the

Tevatron is briefly described below.

FERMILAB'S ACCELERATOR C H A IN

MAIN INJECTOR

RECYCLERTEVATRON

DZERO TARGET HALL

ANTIPROTONSO U R C E

COE

COCKCROFT-WAITON

PROTON

M E SO NNEU TRINO

Figure 2.1: The Fermi National Accelerator Laboratory.

Proton production starts by ionising hydrogen gas to H~ ions which are acceler­

ated to 750 KeV in the Cockcroft-Walton preaccelerator. Further acceleration to

400 MeV takes place in transit through the Linac to the 150 m diameter Booster

synchrotron where the H~ ions are stripped of their electrons and accelerated to

8 GeV. The protons then go to the 1 km diameter Main Injector.

48

Antiprotons produced by colliding 120 GeV protons from the Main Injector onto

a nickel target are captured and stored in the Debuncher and Accumulator re­

spectively, in the triangular Antiproton Source tunnel. They then pass to the

Recycler ring for storage before passing into the Main Injector. For a Tevatron

injection, protons and antiprotons are further accelerated to 150 GeV.

* Initial Instantaneous luminosity at start of store

— Initial Instantaneous luminosity averaged over 20 stores% 10

Jul-02 Jan-03 Jul-03 Jul-04

Figure 2.2: Initial instantaneous luminosity delivered for data used in this anal­ysis.

2.2.1 Tevatron

The Tevatron accelerates protons and antiprotons in a 6 km circular orbit to a

centre-of-mass energy of 1.96 TeV. The protons are kept on a circular orbit by 774

superconducting dipole magnets and focused by 216 superconducting quadrapole

magnets. Eight radio frequency cavities accelerate the particles longitudinally

along the beampipe. The protons and antiprotons are in 36 bunches initially con­

taining approximately 300 x 109 and 50 x 109 particles respectively. These bunches

cross every 396 ns and initially deliver an instantaneous luminosity of approxi­

mately 5 x 1031 cm-2s-1 , providing around 2 interactions per bunch-crossing [27].

The luminosity decreases as the number of antiprotons decreases and, typically

after a day, the collisions are stopped to begin a new ‘store’, replacing the proton

and antiprotons with those produced during the previous store. The instanta­

neous luminosity at the beginning of the stores used in this analysis is shown in

figure 2.2.

2.3 Collider detector at Fermilab

CDF, shown in figure 2.3, is a general-purpose detector designed for high-resolution

kinematic measurements and particle identification. This is achieved using track­

ing, calorimetry and timing information.

The tracking subdetectors, described in section 2.4, provide vertex reconstruc­

tion and momentum measurements for ionising particles using silicon and gas

drift chamber detectors embedded in a 1.4 Tesla magnetic field. Tracking is used

in this analysis for lepton identification and W boson transverse mass recon­

struction, both described in chapter 4. The calorimeter subdetectors, described

in section 2.5, provide an energy measurement of particles that lose energy via

electromagnetic and strong interactions. The calorimeter energy measurement

is used in this analysis for lepton identification and W boson transverse mass

reconstruction in the electron decay channel, both described in chapter 4. Muons

pass through the calorimeters depositing only a small fraction of their total en­

ergy, and are detected in the muon tracking chambers described in section 2.6.

The muon chambers are used in this analysis for muon identification described

in section 4.2. Additional timing information is provided by the Time-of-Flight

detector placed between the solenoid and the tracking volume to improve dis-

50

Figure 2.3: Elevation view of half of the CDF detector.

crimination between low momentum hadrons [30]. The luminosity is measured by

gaseous Cherenkov detectors located close to the beampipe by measuring particles

from inelastic collisions [31]. The miniplug calorimeters located at 3.6 < |?7 | <

5.1 provide information for diffractive physics [32].

2.4 Tracking

Tracking is essential for charged particle identification and reconstruction. It

provides the momentum measurement from the track curvature, which is then

combined with information from regions near the extrapolated track in other

51

subdetectors. Tracks extrapolated to the electromagnetic calorimeter energy de­

posits, or to the muon drift chambers, form the basis of electron and muon recon­

struction respectively. The tracking volume shown in figure 2.4 is embedded in

a uniform (less than 0.1% variation) 1.4 Tesla magnetic field which is parallel to

the beampipe and produced by a superconducting solenoid within a cylindrical

volume 4.8 m long and 3 m in diameter.

Figure 2.4: Tracking volume of the CDF detector.

52

2.4.1 Silicon tracking

The silicon tracking (SVX) detector [33] allows very fine granularity tracking close

to the interaction point in the region | rj \ < 2.0. It comprises three cylindrical

sections 29 cm long. These extend radially from 2.5 cm to 11 cm, providing

high resolution 3-dimensional tracking to reconstruct secondary vertices, includ­

ing bottom quark decays used in top quark searches and bottom quark physics.

Although not required by this analysis, when the SVX is operational it is used

to provide a measurement of the track impact parameter to improve the back­

ground rejection for muon candidate events described in section 4.2. The SVX

is also used to provide data for the beam position measurement used in electron

and muon track reconstruction, described in section 3.1. Additional silicon, not

used in this analysis, is added between the beampipe and the SVX (Layer 00)

to improve vertex reconstruction, and between the SVX and the central outer

tracker (ISL) to improve tracking resolution in the forward regions.

2.4.2 Central outer tracker

The central outer tracker (COT) [34] is a large cylindrical drift chamber used to

track charged particles with \ rj \ < 1 .0 . ThepT measurement used in this analysis

is made using a track reconstructed from COT hits and the beam-position. The

beam-position is measured using SVX hits, if available, or else COT hits. The

COT is located between the SVX and the central calorimeter between 40 cm and

137 cm radially. It extends 155 cm in both directions parallel to the beam pipe

with a spacer to support the field sheets between the two halves at 2 = 0 [35].

The gas mixture and cell arrangement provide a maximum drift time of 100 ns,

53

considerably less than the bunch crossing time of 396 ns.

The COT is constructed from cells 310 cm long, containing 12 wires with an 8

mm separation. The electrostatic field around the sense wires is maintained and

shaped by potential wires, shaper wires and field panels, shown in figure 2.5.

Figure 2.5: The central outer tracker cell layout in superlayer (SL) 2.

The cells are arranged into 8 superlayers (SL) shown in figure 2.6. The superlayers

alternate between the stereo configuration, where the wires make a 3° angle to

the magnetic field, and an axial configuration where the wires are parallel with

the field, providing a position measurement in the z direction.

54

Amplifier-Shaper-Discriminator (ASD) chips are mounted on the cell end-plate

and perform analogue processing on the signal from the sense wire. The sig­

nal is then converted into a digital signal by Time to Digital Converter (TDC)

modules. Information is fed into the Extra Fast Tracker (XFT), described in

section 2.8.1, and the LI Storage Pipeline, the first stage of the data acquisition

system described in section 2.7.

Figure 2.6: Central outer tracker superlayers. Even numbered layers are axial, odd are stereo.

The COT has over 6,000 associated calibration values. Many of these are regularly

updated to maintain high precision data, providing a higher pT-resolution for

leptons used in this analysis. The calibration values include the individual readout

times of the 30240 channels. The readout times are regularly measured by sending

55

a charge pulse to all ASDs simultaneously and comparing the TDC readouts. This

provides each channel with a time offset (to) from the average time. Also fits are

performed whilst data are being taken in order to determine the drift velocity

and the global timing offset.

2.5 Calorimetry

CDF has two types of calorimeter, electromagnetic and hadronic, measuring the

energy of particles that interact via the electromagnetic and strong force respec­

tively. Sampling calorimeters are used, and the photons in the scintillation layers

are reflected to the read-out edge where they traverse a wavelength-shifting light

guide to the photomultiplier tubes (PMT) located on the outer calorimeter edge.

The charge from the PMTs is integrated and converted to a digital signal which

is stored in the LI Storage Pipeline.

The central electromagnetic and hadronic calorimeter subsystems are embedded

in wedges covering 15° in azimuth and arranged in two cylinders, with a small

gap between them at z = 0, to provide complete azimuthal coverage for \rj\ < 1.

The radial location is between the COT and the muon chambers. Each wedge

is divided along its length, in z, into 10 towers that project from the origin and,

except for the last wedge, have equal coverage in pseudorapidity as shown in

figure 2.7.

56

Yi

ught'.'.-•'G uides

M

Lead M iSrirfitetor ' j l LSandwich-^

Chamber

Figure 2.7: Central calorimeter wedge showing the electromagnetic calorimeter towers.

2.5.1 Central electrom agnetic calorimeter

The central electromagnetic (CEM) calorimeter [36] is used in the identification

and energy measurement of electrons and photons. It has 31 layers of scintillator

separated by lead absorbers. Each tower subtends 15° in azimuth and 0.1 units of

pseudorapidity. The measured energy of the high energy electrons and photons

used in this analysis fluctuates about their true energy. The detector resolution,

defined here as the mean fluctuation as a fraction of the measured energy, has

a stochastic component, determined from test beam data as 13.5% /V ^ r, and a

constant component k = 1.5% which are added in quadrature [37]. The detector

response, defined as the ratio between the measured and true energy, together

57

with the resolution are calibrated initially using test beam data and during ac­

celerator shutdowns using a radioactive source. Additionally, a laser system is

regularly used to calibrate changes in response across towers (tower-to-tower cor­

rections). High p t electron tracks also provide time-dependent corrections to the

response. The measurement of the constant term (k) and additional corrections

to the response is described in section 6.4.3.

Additional detectors measuring the 2-dimensional charge distribution are located

at the front of the calorimeter, the central pre-radiator (CPR) detector, and after

the eighth lead layer, the central shower-maximum (CES) detector, where the

shower is at a depth where its transverse size is typically at a maximum.

The CPR and CES consist of gas-based detectors to measure the charge deposition

on orthogonal strips, in the 4> direction, and wires in the z direction. The charge

is integrated and converted to a digital signal which is stored in the LI Storage

Pipeline. Energy deposited in the CES is clustered to determine the location

of the shower centre, in the plane of the CES detector, which is used in this

analysis to select electrons that traverse the central region of the calorimeter

wedge, described in section 4.1.

2.5.2 Central hadronic calorimeter

The central hadronic (CHA) calorimeter [38] has alternating iron and scintillator

layers. The response and resolution of the detector is calibrated using pion test

beams and in situ using isolated tracks. The resolution (aH) for charged pions is

found to be o^ / E t — 50% /yjE t © 3% [37]. The CHA is used in this analysis to

reject events with energy deposits inconsistent with electrons or muons.

58

ENO WALLHADRONCALORIMETER

CRYOSTAT

CENTRAL | TRACK I NO Jf

HADRON CALORIMETER

Figure 2.8: Plug calorimeters.

2.5.3 P lug calorimeters

The ‘plug’ electromagnetic (PEM) and hadronic (PHA) calorimeters, shown in

figure 2.8, cover a range of 1.1 < \g\ < 3.6 and are constructed in a similar fashion

to the central calorimeters with lead absorber between the scintillation layers in

the PEM and steel absorber in the PHA. Located to provide coverage in the gap

between the CHA and the PHA, the ‘end-wall’ (WHA) hadronic calorimeter has

the same structure as the CHA. These detectors are used in this analysis in the

recoil measurement, described in section 4.3, and the missing transverse energy

(# r) measurement described in section 4.4.1.

59

2.6 M uon chambers

The large rest mass of the muon suppresses interaction with m atter via bremsstrahlung

radiation compared to the electron. The muon therefore predominantly interacts

via ionisation, passing through the calorimeters depositing only a small fraction

of its total energy. Muon identification is made using drift chambers and scintil­

lation counters located behind the calorimeters. A muon signature, referred to as

a muon ‘stub’, has hits in multiple drift chambers with scintillation counter hits

synchronous with the beam crossing. In addition to a stub, a muon candidate

has a matching COT track used to measure the muon candidate momentum.

The complete identification requirements for muon candidates are described in

section 4.2.

Electrons and the majority of hadrons are absorbed in the electromagnetic and

hadronic calorimeters and do not reach the muon chambers. However some pions

do make it through the hadronic calorimeter and provide a source of background

in the central muon detector referred to as ‘punch through’. Tau leptons have

an insufficient lifetime to reach the muon chambers. The coverage of the muon

chambers is shown in figure 2.9.

A drift chamber contains a sense wire lying on the same plane as the beampipe,

and is filled with an Argon-Ethane mixture. The chamber is maintained at a high

electrostatic potential by the central wire and the trail of ionisation produced by

a charged particle traversing the gas will drift at a constant speed to the sense

wire. The resultant pulse is then converted into a digital signal by TDCs.

In addition to the three muon subsystems described in more detail below, the bar­

rel muon detector extends coverage to forward regions, covering 1.0 < |r/| < 1.5,

60

Figure 2.9: Muon subdetector coverage in the r\-f> plane (shaded areas). The acceptance of the central muon (CMU) and central muon upgrade (CMP) subde­tectors, located in the region |?7| < 0.65, and the central muon extension (CMX) subdetector, located in the region 0.65 < |?7 | < 1.0, are shown. In this analysis, muons with tracks in the CMU subdetector must also have tracks in the CMP subdetector, and the combined acceptance is determined by the CMP acceptance (shaded), which covers a smaller area in the ij-(J) plane than the CMU.

and is not used in this analysis.

2.6.1 Central muon detector

The central muon (CMU) detector is cylindrical about the beampipe and located

behind the central calorimeter. It consists of 3 modules 226 cm long arranged

directly behind the calorimeter tower to cover 12.6° in azimuth with a 2.4° gap

between wedges. This provides a coverage of \rj\ < 0.65 with a gap of 18 cm

between the two halves for the detector support structure. Each module contains

16 cells shown in figure 2.10. Each cell is 6.5 x 2.5 x 226 cm and has a 50 fiin

stainless steel wire at the centre. The wires of the first and third cells are slightly

offset from the second and fourth and have their wires connected to reduce the

number of channels read out by the TDCs. Similarly, the second and fourth

wires are also connected before being read. The separate pulses traversing the

connected wires are fully resolvable so no loss of information results from this

arrangement. A comparison of drift times across radially aligned wires gives a

crude momentum measurement used for low level triggering. Additionally each

wire has an ADC on each end to determine the z location of the hit.

Muon track i Radial centerline

•55 mm

To pp interaction vertex

Figure 2.10: Transverse view of a CMU wedge.

62

2.6.2 Central muon upgrade

The central muon upgrade (CMP) detector comprises drift chambers with a 2.5 x

15 cm cross-section and typically 640 cm in length arranged in a rectangular ge­

ometry around the CMU, providing approximately the same coverage. It is placed

behind 60 cm of steel greatly reducing charged pion ‘punch through’. Again the

chambers are arranged 4 deep and alternating layers are offset. Additionally,

there is a layer of scintillation counters on the outside of the drift chambers to

provide additional timing information. Muon candidates are required to have

both CMU and CMP hits in addition to scintillation hits that are in time with

the beam crossing.

2.6.3 Central muon extension

The central muon extension (CMX) detector consists of a conical arrangement of

drift chambers about the beampipe in the region 0.65 < \r)\ < 1.0. It covers 240°

in azimuthal angle with a 30° gap at the top for the refrigeration system and a 90°

gap at the bottom where it meets the floor of the collision hall. The drift chambers

differ only in length to those used in the CMP. The trajectory of a particle to

the CMX traverses many more radiation lengths than to the CMU so additional

absorbing material between the drift chambers and the interaction point is not

necessary. A layer of four scintillation counters (CSX) on both the inside and

outside surface of the drift chamber arrangement provides high resolution timing

to reject hits not in time with the collision.

Additional detectors in the ‘keystone’ and ‘miniskirt’ regions were added to fill

the gaps in the original CMX coverage, and cover the top two wedges on the

63

negative z side and the area around the collision hall floor respectively. These

subdetectors are not used in this analysis.

2.7 D ata acquisition

Information from collisions is reduced from a rate of 2.5 MHz, the bunch-crossing

rate, to 75 Hz before being written to tape. A modular system is used to se­

lect events containing signatures of interesting physics, which are subsequently

written to tape based on an analysis using the fully reconstructed event. This

system, shown schematically in figure 2.11, is described below with reference to

the triggers used to select high p r electrons and muons in this analysis.

Physics of interest will have particular signatures that are searched for in three

stages using increasing levels of information and sophistication. The triggering is

split into stages since there is insufficient time for full event reconstruction, which

does not happen unless the third level is reached. Triggers in these stages have

an associated selection efficiency described in section 6.5. Each level has a set of

criteria the data must meet to be written to tape, and the trigger path is the full

set of requirements. About 170 different trigger paths are used at CDF to cover

a broad range of interesting physics. The trigger paths and associated algorithms

used in this analysis are described in section 2.8.

2.7.1 Level 1 trigger

The level 1 trigger (LI) is a synchronous hardware trigger providing a decision

for every event. This is facilitated by having all of the front-end electronics fully

64

Figure 2.11: Trigger dataflow.

pipelined with onboard buffering for 42 beam crossings. The high decision rate

is achieved by fast and simple algorithms that consider low-level objects such as

COT and muon chamber hits and individual calorimeter energy deposits. Hits in

axial COT superlayers are linked together to determine the pT and extrapolated

to other detector regions. Tracks matching a calorimeter tower energy deposit

or hits in the muon chambers are considered for electron and muon candidates

respectively. Also calorimeter energy is summed to indicate the presence of high

Et ’jets’ of hadrons, and to obtain the measurement, defined in section 4.4.1,

used to indicate the presence of a neutrino.

2.7.2 Level 2 trigger

The level 2 trigger (L2) receives events from LI at a rate of 20 kHz and provides an

asynchronous decision based on objects built from combining information written

to a buffer as the result of the LI accept decision. Calorimeter towers are clustered

at this stage and rematched with COT tracks.

2.7.3 Level 3 trigger

The level 3 trigger (L3) receives events from L2 at a rate of 300 Hz. Event data

in the different buffers are collected by the event builder [39] and sent to a farm

of Linux PCs where the events are fully reconstructed. Events take approxi­

mately one second to be processed and those passing the L3 requirements are

subsequently written to robotic tape storage as raw data.

2.7.4 Offline reconstruction

Raw data written to tape are processed again offline, using calibration and align­

ment parameters obtained from the data, before being used for analyses [40]. Of­

fline event reconstruction is done using a farm of Linux PCs, and reconstructed

events are written to tape. The detector calibration and alignment parameters

used for the offline reconstruction are described in chapter 3.

66

2.8 Lepton trigger paths and algorithms

The full set of requirements and the associated algorithms for the selection of

events containing high pt leptons are described below, with a detailed description

of the relevant quantities given in chapter 4.

2.8.1 Extra fast tracker and extrapolation unit

The extra fast tracker (XFT) is used to determine the trajectory and momentum

of tracks in the COT. It is used for online event selection in the LI and L2

triggers as there is insufficient time for full track reconstruction. The first stage,

the finder, takes information from the four axial superlayers and searches for

high Pt segments by comparing hit configurations with pre-programmed look-up

tables. The linker then performs a parallel search on all the segments to find

the highest Pt track made from all four axial superlayer segments in a phi-slice

of A (f) = 1.25°. It has recently been upgraded to accommodate the effect of the

increased chamber occupancy due to the increase in luminosity [41]. Information

then passes to the extrapolation unit (XTRP) and tracks are extrapolated to

other subdetectors.

2.8.2 Electron clustering

At LI, only energy deposits in individual towers are considered. At L2, the central

calorimeter towers are grouped into a 24 x 24 grid in the 7)-<\> plane. Clusters

are then formed recursively by adding adjacent towers that contain sufficient E T

deposits to the seed tower. At L3, the central calorimeter clusters are formed by

67

recursively taking the highest ET tower and adding unclustered adjacent towers

from the same wedge that contain sufficient E t deposits.

2.8.3 Electron trigger paths

An electron candidate passing the electron trigger path is required to have a

COT track that extrapolates to energy deposited in the CEM calorimeter (Eem)

with minimal energy flow (Fhad) into the hadronic calorimeter behind. The

latter requirement is met if the quantity Fjiad/^em Is small- At L3 tracks are

fully reconstructed while LI and L2 use the XFT and XTRP, described in sec­

tion 2.8.1, to determine the track pt and the extrapolated calorimeter position

respectively. The trigger path for high pT electrons used in this analysis is

ELECTR0N_CENTRAL_18.

The requirements of the ELECTR0N_CENTRAL_18 trigger path are:

L1_CEM8_PT8 at LI which requires two CEM towers within the same wedge with

a combined ET greater than 8 GeV and Ehad/^em less than 0.125 and an

XFT track that extrapolates to the tower. The XFT track must have at

least 10 (11) hits in at least 3 (4) superlayers with pt greater than 8.34

GeV.

L2_CEM16_PT8 at L2 which requires a central EM cluster with Et greater than

16 GeV and Fhad/-Fem less than 0.125 and an XFT track with pt greater

than 8.34 GeV that extrapolates to the cluster seed tower.

L3_ELECTR0N_CENTRAL_18 at L3 which requires a reconstructed cluster with ET

greater than 18 GeV and FWi/Fem less than 0.125 and a COT track with

68

P t greater than 9 GeV that extrapolates to the cluster. In addition, from

January 2003 the following four requirements are made: Firstly, the com­

parison of the energy profile of the towers in a cluster with that predicted

from test beam data, defined in equation 4.1, is required to be less than 0.4.

Secondly, the distance in the z direction between the track CES position

and the matched cluster position in the CES is required to be less than 8

cm. Thirdly, the E t is calculated using the origin of the extrapolated track

instead of the nominal interaction point and lastly the number of hadronic

towers used in the calculation of E hsud/E em is increased from two to three.

The WJIOTRACK trigger path selects electron candidates from W boson decays

without any track requirements. The presence of the W boson is inferred from

large P t carried away by the neutrino. This trigger path is used to measure the

efficiencies of the XFT requirements in the ELECTR0N_CENTRAL_18 trigger path.

The requirements of the WJJOTRACK trigger path are:

L1_EM8_&_MET15 at LI which requires, in addition to the calorimeter require­

ments in L1_CEM8_PT8, a P t greater than 15 GeV. From January 2003, the

plug calorimeter is included in the P t sum.

L2_CEM16_L1_MET15 at L2 which requires, in addition to only the calorimeter

requirements of L2_CEM16_PT8, the LI p T requirement.

L3_W_N0TRACK_MET25 at L3 which requires, in addition to the calorimeter re­

quirements of L3_ELECTR0N_CENTRAL_18, a P t greater than 25 GeV.

69

2.8.4 M uon trigger paths

A muon candidate passing the muon trigger is required to have a stub in the muon

drift chambers with a matching COT track. A stub requires multiple matching

hits in either the CMU and CMP chambers, or the CMX chambers and the CSX

scintillators. For LI and L2, the track px and extrapolated position in the muon

chamber is determined by the XFT and the XTRP respectively. At L3 the COT

track is fully reconstructed and extrapolated to the muon chambers to calculate

the distance (A:rcmu, Axcmp and A x cmx) between the extrapolated track and the

stub in the CMU, CMP and CMX chamber respectively.

The trigger paths for high pT muons used in this analysis are MU0N_CMUP18 and

MU0N_CMX18. An increasingly stringent L2 trigger requirement was required to

keep the rate of other ‘fake’ muon events low as the luminosity increases.

The requirements of the MU0N_CMUP18 trigger path are:

L1_CMUP6JPT4 at LI which requires hits on both CMU wire pairs within 124 ns

of each other. These hits must have a matching XFT track with px greater

than 4.09 GeV. The CMP must have hits in 3 of 4 layers in the projected

direction of the CMU hits.

L2_AUT0_L1_CMUP6_PT4 at L2 which requires the corresponding LI trigger. This

trigger was used until October 2002.

L2_TRK8_L1_CMUP6_PT4 at L2 which requires an XFT track with px greater than

8.34 GeV. This trigger was used between October 2002 and April 2004.

L2_CMUP6_PT8 at L2 which requires an XFT track with px greater than 8.34

70

GeV extrapolating to hits in the CMU and CMP. This trigger was used

after April 2004.

L3_MU0N_CMUP18 at L3 which requires a reconstructed muon candidate with a

track pT greater than 18 GeV. The reconstructed track must extrapolate

to a stub in the CMU and CMP with |Axcmu| less than 10 cm and |Axcmp|

less than 20 cm.

The requirements of the MU0N_CMX18 trigger are

L1_CMX6_PT8 at LI which requires hits on both CMX wire pairs within 124 ns

of each other. The hits must have a matching XFT track with pT greater

than 8.34 GeV. This trigger was used until October 2002.

L1_CMX6_PT8_CSX at LI which requires, in addition to the above trigger require­

ments, a CSX scintillator hit. This trigger was used after October 2002.

L2_AUT0_L1_CMX6_PT8 and L2_AUT0_L1_CMX6_PT8_CSX at L2 which requires the

corresponding LI trigger. This trigger was used until April 2004.

L2_CMX6_PT10 at L2 which requires an XFT track with a pt greater than 10.1

GeV extrapolating to hits in the CMX. This trigger was used after April

2004.

L3_MU0N_CMX18 at L3 which requires a reconstructed muon with a track Pt of at

least 18 GeV. The reconstructed track must link with the CMX stub with

|Axcmx| less than 10 cm.

71

2.9 D atasets

Data written to tape is catalogued into different datasets depending on the type

of event, as determined by the associated trigger paths described in section 2.8.

Events of interest for this analysis will contain high pT electrons or muons.

The high pT electron dataset containing electrons from the ELECTR0N_CENTRAL_18

and WJJOTRACK trigger path is used in this analysis. The W_N0TRACK trigger path

is used to estimate the efficiency of the ELECTR0N_CENTRAL_18 trigger path in

section 6.5.1 and the ELECTR0N_CENTRAL_18 trigger path is used for all other

electrons. The high pT muon dataset containing muons from the MU0N_CMUP18

and MU0N_CMX18 trigger paths is used for muons. Only data collected when the

necessary subdetectors are fully operational are used in this analysis, and are

selected according to the standard ‘good ru n ’ criteria [42], but without requiring

the silicon subdetector to be functional. This gives an integrated luminosity of

370 ± 18 pb-1 for electrons and 330 ± 16 pb-1 for muons.

72

Chapter 3

Calibration m anagem ent

Each sub-system of the CDF detector described in chapter 2 has components that

are calibrated, often regularly. The calibration values, such as those produced

during CEM calibrations described in section 2.5.1 and values describing the sta­

tus of the detector, are stored in a relational database for retrieval at a later date.

These individual values, henceforth referred to as calibrations, are collected into a

single table and accessed when reconstructing events offline for subsequent use in

analyses. This chapter describes the process in detail with reference to the central

electromagnetic calorimeters and COT calibrations, described in section 2.5.1 and

2.4.2 respectively, as these are vital in obtaining a good resolution for the energy

and momentum measurement, described in section 6.4 and 6.3 respectively, and

used in the fit described in chapter 8 to determine the W boson decay width.

The author’s contribution to this area is the creation of a set of tools that bring the

individual calibrations together into a single complete set, described in section 3.3,

and the monitoring and validation of this process, described in section 3.4.

73

3.1 Types of calibrations

The calibrations fall into three broad groups based on the method of production

and the timescale over which they change. Data written to tape are grouped into

sections called ‘runs’, typically lasting a few hours, which represent a collection

of data for which the detector hardware and software configurations remained

unchanged. This is the shortest timescale over which calibrations change.

Online calibrations depend on the exact running conditions that prevail whilst

data are being collected. These are written to the database for each run

whilst the run is in progress. Examples include the COT drift model fits

and general detector parameters such as the operating voltage and list of

bad channels.

Offline calibrations change over longer time scales and are made by performing

special detector calibrations typically every few months. Examples include

the CEM time dependent gain calibrations described in section 2.5.1.

Post reconstruction calibrations are produced from the analysis of fully re­

constructed events. These are made before the production of the full offline

reconstruction of the events used for analyses. Examples include the beam

position produced from reconstructed vertices, that are used in this analysis

to provide an extra point when fitting to track hits, described in section 4.1.

74

3.2 Database structure

Each run is associated with a complete set of calibrations comprising over 150,000

individual values used for the offline reconstruction of events. As the detector

system is inherently modular, with each subdetector producing its own set of

calibrations either online or later offline, many intermediate calibration sets are

created before the final set. Also by the time the CDF detector stops taking

data, the number of runs is likely to be in excess of 100,000 so minimising the

database size is an essential consideration of database design. Additionally some

of the calibrations are unchanged over multiple runs and final calibration sets for

each run contain many identical elements. Database size is kept to a minimum

by using a relational database structure where each individual calibration only

has a single entry.

Simple access to the complete set of ‘official’ calibrations is necessary for offline

reconstruction and other uses. Also a record of all database activity is necessary

to eliminate possible data loss and to allow complete access to all past conditions.

This is achieved by disallowing an entry to be changed once written. Another

important consideration is the ability to merge two sets of calibrations that both

contain instances of the same calibration, taking only the latest. The schema

used, based on these considerations, is described below. An Oracle database is

used, and is interfaced to using the Structured Query Language (SQL) [43].

3.2.1 Calibration retrieval

Simple access to a set of calibrations for a range of runs (defined as a ‘pass’) must

be possible without a detailed knowledge of the database structure. A simple

75

CALIBRUNLISTS:

CALIB TABL* CID(COTSTAGEO) C594601]

S E T R U N N IAPS:

CID / JOBSET

"594601) 1604582631624 1604582560411 1604582594543 1604582594733 1604582

(COTSTAGEO 0

CID PARNUM SL1 SL2 ...

594601 1 -.08 -.075594601 2 .154197 .238434594601 3 190.102077533727 190.46304162641594601 4 -12.4854450076531 -12.5272084462342594601 5 .3 .3594601 6 0 0594601 7 .05 .05594601 8 243.457264676525 263.077426187181594601 9 0 0594601 10 0 0594601 11 0 0594601 12 0 0

USED SETS;JOBSET PROCESS NAME

isPROCESS RUN PROC CALIB VERSION

16^6458 PROD PHYSICS CDF 241664 1604582)<fRQD~PHYSICS CDg) (241664)

PASSES:PASSNAME PASS INDEX PROCESS_NJWE RETIRED j..17 179 PROD PHYSICS CDF YES I17 ) (180^ (PROD PHYSICS CDg)NULL I

[ pass name j (run number]

C 241664)<.2

PA SSC A LIBS:

PASS INDEX LORUN PRtJC CALIB VERSION

Figure 3.1: Calibration database schema showing the relevant tables to retrieve the calibrations from the COTSTAGEO table as an example.

76

interface to a set of calibrations for a given run is provided, with the user only

specifying the run number and the pass name of the required calibration set.

The database schema linking a run number and pass name to a unique set of

calibrations is shown in figure 3.1, using the COTSTAGEO table, which contains

the drift fit values for the COT calibrations described in section 2.4.2, as an

example. Only table fields relevant to the schema structure are shown. The five

intermediate tables used for the retrieval of the COTSTAGEO calibration table, one

of the 104 calibration tables identified by a given pass name, are described below.

PASSES associates a given PASSNAME with a PROCESSJIAME (PROD_PHYSICS_CDF)

and a pass number, the PASS_INDEX. Each PASSNAME defines a pass and, as

it may be associated with more than one PASS.INDEX value, has a RETIRED

field set to ‘YES’ for all passes except the current pass, which has the de­

fault ‘NULL’ value. The PASSNAME requested by the general user (in this

example ‘17’) uniquely identifies the PASS_INDEX (180) for the current pass.

It is sometimes necessary to update the calibrations in a pass and a new

PASS.INDEX for a given PASSNAME is created by updating the RETIRED field

to ‘YES’ in the current row and creating a new row.

PASSCALIBS associates a given run number (241664), the LORUN, with a version

number (2), the PR0C_CALIB_VERSI0N, for the PASS.INDEX specified by the

PASSES table.

USED.SETS associates a JOBSET number (1604582) with the PROCESSJIAME given

by the PASSES table, and the LORUN and PR0C_CALIB_VERSI0N given by the

PASSCALIBS table.

SETJRUNJ1APS associates the JOBSET with a set of CID numbers, each identify­

77

ing an individual calibration table. This allows the grouping of individual

calibration tables by using a common JOBSET number.

CALIBRUNLISTS associates the CID (594601) with the table name (COTSTAGEO), in

the CALIB_TABLE field. It is often necessary to combine two sets of calibra­

tion tables, both grouped by a common JOBSET number in the SET_RUN_MAPS

table, into a new grouping with a new common JOBSET number. The

CREATE_DATE field (not shown), allows the most recent instance of two iden­

tical CALIB.TABLE names to be included in the new combined set.

COTSTAGEO and all the other tables containing calibration values are identified

by the CALIB-TABLE name. The CID specifies a specific instance of the type

of calibrations stored in the table. For COTSTAGEO, each row contains one

of the 12 parameters associated with the drift model for each of the 8 COT

superlayers described in section 2.4.2.

3.3 Offline procedure for calibration

For each run the calibration CIDs are grouped by a common JOBSET number

in the SET_RUN_MAPS table, with the PROCESSJJAME of PROD_PHYSICS_CDF, in the

USED_SETS table. As the production of reconstructed events for analyses rely

on these calibrations, a robust procedure that is automated where possible is

essential to ensure efficient availability of reconstructed data. The calibrations

are merged at different stages into intermediate passes of PROD_PHYSICS_CDF.

The pass evolution, with the calibrations relevant to this analysis, are shown in

figure 3.2.

78

COT calibrations

COT beamlines

Trigger information and detector status

Calorimetercalibrations * \K►( Initial Pass

Intermediate Pass

Beamlines Pass

Calorimetercalibrations KH Test Final Pass

SVX calibrations

SVX beamlines

CFinal Pass

Figure 3.2: Pass evolution with calibrations used in this analysis.

A key aspect of a robust system is a modular approach and the availability of

software tools. Automated and helper tools for creating and monitoring calibra­

tions are built from generic modular functions that interface with the database.

79

This provides an interface that is easily adapted to changing database schema and

procedures, and greatly reduces software based errors, and testing and validation

time.

Due to the different types of calibrations described in section 3.1 and their pro­

duction timescale, several intermediate passes are made before the set is complete.

To ensure efficient availability, the intermediate stages are created automatically

when the necessary tables become available, independent of the status of other

runs.

The first version of PROD_PHYSICS_CDF is made from merging all available cali­

brations, such as those from the trigger system, when the run ends. This is put

into the ‘initial pass’ which is used for preliminary studies.

The calibrations for the tracking subdetectors described in section 2.4 are pro­

duced using multiple runs. They are merged and put into the ‘intermediate pass’

when they become available, typically within 24 hours of the run ending. The

beam positions, described in section 3.1, are produced once the intermediate pass

is available. They are merged and put into the ‘beamlines pass’, typically within

36 hours of the run ending.

The ‘final pass’ is used for the full offline production of data for analyses. The

tables added at this stage are made from analysing reconstructed data collected

over the previous few months such as the CEM calorimeter calibrations described

in section 2.5.1. Before the final pass is created, a ‘test pass’ is created and checked

extensively over a period of a few weeks.

80

3.4 M onitoring and validating calibrations

Monitoring the status of calibrations is an essential aspect of calibration produc­

tion. This is done automatically until the final pass is available. For passes not

available within the specified timescales, the presence of the necessary tables is

checked and the relevant expert emailed.

For manual validation of calibrations, a tool to compare channels in different

JOBSETs is used. Details of channels common to both JOBSETs with different CID

numbers are provided, optionally including channels present in only one JOBSET.

In addition to individual JOBSETs, comparisons between a pass and either another

pass or the latest version are also possible. This will quickly confirm whether

a newly created pass contains the most recent calibrations or whether certain

calibrations have been successfully updated leaving the others unchanged. As

the pass is unavailable whilst values are being changed, this tool allows the down

time to be kept to a minimum whilst providing rigorous validation.

81

Chapter 4

Event selection and

reconstruction

This measurement of Tw is made in the W —► eu and W (iv decay channels

using data with an integrated luminosity of 370 pb-1 and 330 pb-1 respectively,

collected at CDF between February 2002 and August 2004. The Z —> ee and

Z —► fifi decay channels, where the opposite charge of the lepton pair is implied,

provide a high-purity event sample and are used extensively to calibrate the

simulation. The large W and Z boson masses result in decay products with

large transverse momenta. The presence of a neutrino, whose direct detection

at CDF is not possible, is inferred from missing transverse energy. W boson

candidate events contain a high pT electron or muon and large missing transverse

energy, and Z boson candidate events contain two high pt electrons or muons

with opposite charge. Electron and muon candidates with tracks traversing the

COT are required to extrapolate to the CEM energy deposits and muon chamber

hits respectively.

82

Events at CDF are fully reconstructed offline from raw data using the latest

calibrations described in chapter 3. The selection and reconstruction of objects

used in this analysis is described below.

4.1 Electron selection

Electrons are identified by matching COT tracks with energy deposited in the

CEM using the position provided by the CES. The requirements met by an elec­

tron candidate are detailed below.

Energy reconstruction

Energy deposited by an electron traversing the CEM is not always contained in

a single tower. Similar to the L3 electron clustering, described in section 2.8.2,

CEM energy deposits are recursively grouped by adding the unclustered tower

with highest E t to the unclustered neighbouring towrer in the same wedge with

the highest E T. Energy deposited in the CES, described in section 2.5.1, is also

clustered to provide the shower position measurement.

Calorim eter requirem ents

To ensure the electromagnetic shower is contained in a well instrumented region

of the calorimeter, the shower is required to be in a ‘fiducial’ region. Candidate

electrons with shower leakage into the gap at \z\ < 4.2 cm, where the two halves

of the calorimeter meet, are excluded by requiring the distance (2:ces) in the z

direction between the CES shower position and 2 = 0 to be greater than 12 cm.

83

Additionally, the distance (xces) from the wedge center, perpendicular to the z

direction in the CES plane, is required to be less than 18 cm to minimise shower

Since electrons deposit most of their energy in the electromagnetic calorimeter,

■E'had/Eem of electron candidates is required to be less than 0.07. Electron can­

didates are required to be in the first 9 towers (towers 0 to 8) from z = 0 as

those furthest from the centre on either side have greater energy leakage into the

hadronic calorimeter.

In addition to energy deposited in the tower containing the candidate electron,

energy is also deposited in the two neighbouring towers in the same wedge. This

‘energy profile’ of candidate electrons can be compared to that of electrons from

test beams using the parameterisation [44]

where E{ is the measured tower energy in the neighbouring tower, ^ xpected is the

expected energy in that tower, measured from test beam data, and Ai^expected

less than 0.3 are considered consistent with electrons and therefore required for

electron candidates.

leakage into the gap where the calorimeter wedges meet 23.1 cm from the wedge

centre.

y j (0.14\/Ei)'2 + ( A £ f pccted)2

cluster towers i

(4.1)

is the measured fluctuation of the latter. Electromagnetic clusters with Lshr

84

Track requirem ents

Individual COT hits and the beam position, obtained from reconstructed vertices,

are fitted with a helix to provide a ‘beam constrained’ track. The number of axial

and stereo COT superlayers with at least 7 hits (TV3*1*11 and N stereo respectively) is

required to be at least 3. The pt of the beam constrained track is calculated from

the helix curvature and the magnetic field strength, and is required to be greater

than 9 GeV. The track impact parameter in the z direction (z0) is required to be

less than 60 cm from the detector center. Neglecting the electron rest mass, the

true P t and E t are equal, so electron candidates are required to have a cluster ET

between 0.8 and 1.3 times the track pT, although this requirement is relaxed in

studies to measure parameters for the detector simulation described in chapter 6.

Tracks are extrapolated to the plane of the CES to match a track with a cluster.

The CES track position is required to be less than 8 cm from the CES cluster

position in the z direction (Az), and less than 10 cm in the perpendicular direction

in the CES plane (Ax).

The full requirements of electron candidates are summarised in table 4.1.

4.2 M uon selection

An event containing a muon is identified by a track in the muon drift chamber

matching a track in the COT, and energy deposits in the calorimeter consistent

with a muon.

85

E'y > 25 GeVPt > 9 GeVN < 60 cml ces | < 18 cm| ces | > 12 cmE t /P t > 0.8 and < 1.3E\iaxl/ E e m < 0.07Tshr < 0.3\Az\ < 8 cm|Ax| < 10 cmyy axial > 3 SL with > 7 hits/SLyystereo > 3 SL with > 7 hits/SL

Table 4.1: Requirements for electron candidates.

Track requirem ents

The total energy of the muon cannot be measured as the muon passes through

the entire detector without coming to rest. The muon pt is determined from the

curvature of the helix fit to the COT hits and the beam position. Each muon

candidate track is required to have at least 7 hits in 3 axial and 3 stereo COT

superlayers. To allow an accurate measurement of the track impact parameter

in the xy plane (do), hits are required in the silicon detector when operational

otherwise the number of hits in the first COT superlayer is required to

be at least 5. To ensure the track traverses all 8 superlayers, the radius at which

the track crosses the COT end-plate (rcot) is required to be greater than 137

cm, the outer radius of the COT. The track impact parameter in the z direction

is required to be less than 60 cm from the detector center and the track do is

required to be less than 0.04 cm from the detector center for tracks with silicon

hits and less then 0.2 cm for those without. Candidate muon track are required

to have a good helix fit by demanding the x^rack/ndf to be less than 3 for tracks

with silicon and less than 2 for tracks without, where ndf = N ^ \s — 5 where 5 is

86

the number of free parameters in the helix fit.

Cosmic muon rejection

High energy ‘cosmic’ muons from space are constantly passing through the CDF

detector, and are excluded from candidate muon events. Many of these cosmic

muons meet the muon selection requirements, and can either mimic a Z boson

event if both the incoming and outgoing tracks are reconstructed or a W boson

event if only the incoming or outgoing track is reconstructed. The cosmic muon

tagging algorithm performs a helix fit to the track hits with a floating global

time offset for all hits included. The hits are fitted twice, with an outgoing

and an incoming (time reversed) hypothesis. The fit with the lowest x 2rack/nd f

determines the track direction. Hits in the opposite side of the detector in the

region extrapolated to by the first refitted track are also fitted in the same way

and are included in a single helix fit to both tracks, again with a global time offset.

The colinearity of the two fits is ensured by requiring the azimuthal angle (A(f)^)

between the tracks to be greater than 3.13 and the sum of the track rapidity

values (53 Vnn) t° be less than 0.03. Events with individual fits consistent with a

colinear incoming-outgoing hypothesis and a combined fit with y 2 less then 300

are tagged as cosmic muons and rejected from the candidate events.

Calorim eter requirem ents

To ensure the calorimeter deposits are consistent with a muon, the wedge tra ­

versed by the extrapolated track is required to have less than 2 GeV deposited

in the electromagnetic calorimeter and less than 5.6 + 0.014xpT GeV in the

87

hadronic calorimeter. The latter requirement ensures a pt independent selection

efficiency, described in section 6.5.4.

The track and calorimeter requirements for muon candidates are summarised in

table 4.2.

> 25 GeVkol < 60 cmsilicon hits when silicon is ‘good’jy axial > 3 with > 7 hits/SLjy stereo > 3 with > 7 hits/SLjVhit. > 5 (no silicon tracks only)T'cot > 137 cmdo < 0.04(0.2) cm for silicon(no silicon)E• em < 2 GeV- had < 5.6 + 0.014xpT GeVX?rack/ndf < 3(2) for silicon (no silicon)cosmic not true

Table 4.2: Track and calorimeter requirements for muon candidates.

M uon chamber requirements

In addition to the track and calorimeter requirements described above and listed

in table 4.2, candidate muons have the stub requirements described below. To

ensure the muon passes through a fiducial region of the muon chamber, the helix

of the COT fit is propagated through a simulation of the muon chamber geometry.

This determines the minimum distance from the cell boundaries in the x and z

directions in local chamber coordinates, which correspond to the perpendicular

and parallel directions to the drift wire in the chamber plane respectively. The

muon chamber hits and the extrapolated COT track are in a fiducial region

if they are within the chamber in the local x direction and more than 3 cm

within the chamber in the local z direction. The distance between the COT

track extrapolated to the muon chambers and the reconstructed muon chamber

track is required to be within 3, 6 or 5 cm in the local x direction for the CMU,

CMP and CMX chamber respectively (Axcmu, A2:cmp and Axcmx respectively).

These requirements are listed in table 4.3, where a CMUP stub has hits in both

the CMU and CMP chambers.

CMUP or CMX stub present |Axcmp| < 6 cm|Axcmu| 3 cm|Axcmx| < 5 cmfiducial true

Table 4.3: Stub requirements for muon candidates.

4.3 Recoil reconstruction

In addition to energy deposited by high pt leptons, there are three other sources

of calorimeter activity in candidate W and Z boson events at CDF:

Initial sta te QCD radiation, where quarks and gluons are radiated from the

two partons participating in the hard scatter, described in section 1.2. This

results in calorimeter activity that is strongly correlated with boson pr.

Spectator parton radiation, where remnant partons from the broken inter­

acting pp undergo QCD interactions to produce hadrons. Most of the

energy flow is at low angle and subsequently undetected, and the detected

energy is largely uncorrelated with luminosity and boson p f .

A dditional proton-antiproton (m inim um bias) interactions, where en­

ergy from interactions between other pps in the beam is deposited in the

89

calorimeters. This results in calorimeter activity that is strongly correlated

with luminosity and uniformly distributed in azimuthal angle.

All calorimeter activity not associated with the lepton is collectively referred

to as the recoil, with a ‘hard’ component from QCD radiation from the boson

production, and a ‘soft’ component from the spectator parton interactions and

minimum bias events.

The towers surrounding the lepton tower contain energy from final state photon

radiation and lepton bremsstrahlung radiation, as well as leakage of the electro­

magnetic shower. To exclude energy associated with the lepton from the recoil,

a ‘knockout region’ of towers surrounding the lepton is removed from the recoil

sum. The knockout region is chosen to maximises both the inclusion of energy

associated with the lepton and the exclusion of energy associated with the recoil.

Although the tower with the lepton CES shower position contains most of the de­

posited lepton energy, there is considerable leakage of the electromagnetic shower

into the adjacent tower in the same wedge nearest the CES shower position. For

electrons, these two towers are identified by the electron clustering algorithm and

used in the electron energy reconstruction. However, there is additional shower

leakage into the other neighbouring towers, predominantly into the tower in the

adjacent wedge nearest the shower CES position. The knockout region is there­

fore defined relative to the direction of the two neighbouring towers nearest the

shower CES position, defined as the (f)ces and r]ces directions.

The knockout region, centered on the highest energy tower, is shown in figure 4.1

for candidate electrons and muons, together with the average energy deposited

in each neighbouring tower in the electromagnetic and hadronic calorimeter.

90

Figure 4.1: The average energy deposited in M eV in the tower containing the CES shower position and the neighbouring towers for the (left) electromagnetic and (right) hadronic calorimeter for (top) electron candidates m W —► eis events and (bottom) muon candidates in W —► pv events. Arjces = +1 and A 4>ces = 4-1 denote the nearest tower to the shower CES position in the same wedge and the adjacent wedge respectively. The shaded region is the knockout region.

Since both adjacent towers in the same wedge contain significant energy leakage

they are included in the knockout region. Additionally, for electrons, the three

towers in the adjacent wedge nearest the CES shower position as well as the

neighbouring tower in the opposite adjacent wedge also contain significant leakage

energy, together with wide angle photons radiated by the electron, and are also

included in the electron knockout region.

The recoil energy vector (u) and total transverse energy sum (£_ET) are defined

as the transverse vector and scalar sum respectively of all tower energy deposits

greater than 100 MeV, excluding the miniplug detector. Tower energy deposits

in the knockout region are replaced with an average ‘underlying event’ energy to

include the contribution from the soft ‘minimum bias’ events that underlie the

electron energy deposits.

For W candidate events the recoil components uy and u± are defined along and

perpendicular to the lepton direction respectively, while for Z candidate events

the recoil components U\ and u2 are defined along and perpendicular to the p f

direction respectively. By defining the direction of the recoil components with

respect to the Z boson pT, the hard and soft recoil contributions are partially

decoupled, facilitating the recoil simulation described in section 6.6.

The underlying event energy is estimated by sampling a 7 tower region at the

same p and with A0 = 90° from the lepton knockout region in W —> eis candidate

events. For consistency with the recoil definition, only towers with energy deposits

greater than 100 MeV are included in the average. The average tower energy

is found to be 33 MeV for the electromagnetic calorimeter and 9 MeV for the

hadronic calorimeter for both electron and muon candidates.

The underlying event energy has a hard and a soft contribution. The hard contri­

bution has the dependence on u\\ shown in figure 4.2 (left), being largest when u\\

is largest as the recoil is along the lepton direction. The soft contribution has the

luminosity dependence shown in figure 4.2 (right) as the number of minimum-bias

events increases with luminosity.

92

90)(53Ui

0.35

0.3

0.25

40 60Luminosity x 10'30 (cmV)

% 0.9C30.8«

!u0.7

0.6

0.5

0.4

0.3

0.2

u,| (GeV)

Figure 4.2: Underlying event energy dependence the recoil component parallel to the lepton direction (u\\), and luminosity, in W —> eu candidate events.

% 0.34

tu 0.32

0.3

0.28

0.26

0.24

Figure 4.3: The distribution of the average underlying event energy per tower as a function of lepton tower rj. The energy is measured in a 7 tower region in the electromagnetic calorimeter, with a tower threshold of 100 MeV, i n W - * e v candidate events.

The underlying event energy also has a dependence on lepton rj, shown in fig­

ure 4.3. For the recoil reconstruction, the underlying event energy is added back

into each tower in the knockout region with the u\\ and lepton rj dependence ob­

tained from sampling the distributions in figure 4.2 (left) and 4.3 respectively, and

with the luminosity dependence obtained from the linear fit to the distribution

in figure 4.2 (right).

93

4.4 Boson mass reconstruction

As the leptons used in this analysis to reconstruct the boson mass are highly

relativistic, with a Lorentz factor greater than 240, their rest masses can be

neglected. For muon candidates, the track momentum is used to calculate the

invariant mass. For electron candidates the calorimeter energy (E) is used, as

it is known with greater accuracy than the track momentum, with the direction

determined from the CES shower position and the track vertex.

4.4.1 W boson rax reconstruction

The neutrino is not detected at CDF so its momentum cannot be directly re­

constructed. Since there is energy flow, particularly from spectator parton QCD

interactions, through uninstrumented regions at large rapidity, the only estimate

of the neutrino momentum is the missing energy in the transverse plane ($ t ),

defined as

-^ t — —{Et + u) (4-2)

where u is the recoil vector described in section 4.3. For W —> /iv events, the

muon p r is substituted for E t .

For W boson candidate events, the mass in the transverse plane is reconstructed

with equation 1.21 using for the neutrino transverse momentum and E t ( p t )

for the lepton transverse momentum in W —> eis (W —> nv) events.

94

4.4.2 Z boson mass reconstruction

The invariant mass is calculated using equation 1.20. In Z —► [i/i candidate events

the invariant mass (mw ) is calculated for the two highest Pt muon candidates

using the muon pr- In Z —» ee candidate events, the invariant mass (mee) is

calculated for the two highest Et electron candidates using the electron E t-

4.5 Boson selection

In addition to W boson candidate events selected for the measurement of Tw,

Z bosons candidate events are selected with high purity and with a fully re­

constructed boson mass. This facilitates the tuning of the event and detector

simulation, described in chapters 5 and 6 respectively, and the determination of

the background event distributions described in chapter 7, together with their

associated systematic uncertainties.

Both W and Z boson candidate events are required to have a recoil magnitude

(|w|) less than 20 GeV to reduce QCD background events in W candidate events,

described in section 7.3. However, for Z boson candidates used in the boson

Pt and recoil studies described in sections 5.2 and 6.6 respectively, the recoil

requirement is replaced with a Z boson pT (p^) requirement of less than 50 GeV.

W boson selection

W boson candidate events are single lepton events with greater than 25 GeV

and no additional high pT track with opposite charge. The latter ‘Z veto’ require-

95

ment rejects events that contain a Z boson where one of the leptons traverses a

gap in the calorimeter or muon chamber.

Pt > 20 GeVAz0 < 5 cmCOT hits presentopposite sign true

W - ■* ev candidatesv-~vA /i< 0 .4E P t < 5 GeVI Tees | > 18 cml cesj < 9 cm

w - » pv candidatescosmic not true

andE P t < 10 GeV

orV '' A i? < 0 .4E Pt < 2 GeVFem < 2 GeVF-had < 5.6 + 0.014xpr GeV

Table 4.4: Z veto requirements for additional track.

Z veto events have an additional track with COT hits, a Pt greater than 20

GeV, a distance between zq and that of the candidate track (A 2 0 ) less than 5 cm

and no additional high pT tracks nearby. The latter ‘track isolation’ requirement

is met by demanding the pT sum of all other tracks within a cone of A R =

■sj(Arj)2 + (A(f>)2 = 0.4 with Azo less than 5 cm (%2AR<0AP t ) to be less than 5

GeV for W —> e v candidate events. For W —► f iis candidate events, J 2 A R < 0 A P t

is required to be either less than 10 GeV if the calorimeter energy deposits are

consistent with a muon or less than 2 GeV otherwise. These requirements are

listed in table 4.4 and the requirements for W boson candidate events are listed

in table 4.5.

The ttt-t distributions of the 108,808 W —» ev candidate events and 127,432 W —> /iv

candidate events are used to measure the W boson decay width, described in

96

# T > 25 GeVmx > 50 GeV and < 200 GeV|u| < 20 GeVZ veto false

Table 4.5: W boson requirements.

chapter 8 .

4.5.1 Z boson selection

The yield of Z —> nfi candidate events is increased without significant loss of

purity, by requiring only a single ‘tight’ muon candidate which meets the full

requirements listed in section 4.2. The other ‘loose’ muon candidate is required

to meet only the track and calorimeter requirements listed in table 4.2. The data

contain 6267 candidate Z —*■ /i /i events and 2903 candidate Z —»• ee events that

meet the requirements listed in table 4.6.

~m~u > 80 GeV and < 1 0 0 GeVLepton charge Opposite signRecoil < 20 GeV

Table 4.6: Z boson requirements.

97

Chapter 5

Event generation

The raT distribution of W bosons has a Breit-Wigner component, from the prop­

agator lifetime described in section 1.3, and a Gaussian component from the finite

resolution of the detector. The Gaussian resolution component decreases faster

with raT than the Breit-Wigner component, rendering the high raT region sensi­

tive to Tw , shown in figure 1.11. Thus a measurement of Tw can be extracted

by comparing the simulated raT distribution to data in the high ttit region, after

normalisation in the central region. This method was also used in the previous

CDF measurement of Tw [23].

The simulation of the tu t distribution has three components; the event genera­

tion, described in this chapter, the detector simulation, described in chapter 6

and the simulation of the fraction and raT distribution of ‘background’ processes

that contaminate the candidate events, described in chapter 7.

The estimation of the systematic uncertainties associated with the event simula­

tion, summarised in section 8 .1 , are also described, and are taken as the deviation

98

of the measured value of Tw when the particular simulation method is varied by

an amount representing the la confidence level. This is achieved by generat­

ing two sets of simulated ‘pseudo-data’, one with the standard parameterisation

and the other with a change representing the la deviation. The simulated

distribution is fitted to the two sets of pseudo-data to determine the shift in T w

The simulated m t distribution is fitted to that of candidate events in the region

ra^ < rri’T < 200 GeV, and normalised in the region 50 < mx < mjf GeV. The

value (m??1) of the lower bound of the fit region affects the size of the systematic

and statistical uncertainty associated with the measurement of T w Specifically,

the larger the value of m ^ , the smaller the systematic uncertainty and the larger

the statistical uncertainty. The systematic uncertainties are estimated for ra^

values of 80, 85, 90, 100, 110 GeV. The value of that minimises the combined

systematic and statistical uncertainty is evaluated in section 8 .1 .

The Drell-Yan boson production process, described in section 1.2, has three com­

ponents that need to be simulated; the QCD effects of the interacting partons

(the PDFs and boson p r), the QED radiative effects on the boson and final state

leptons, and the decay kinematics of the boson (including the boson pT and po­

larisation dependence). Since a combined NNLO calculation of the QCD and

QED effects is not available, a leading-order event generator is used with NLO

PDFs and NLO QED radiative corrections. A separate NLO calculation of is

used to simulated the QCD effects resulting in a non-zero boson pT. This chapter

describes the event generation and the associated systematic uncertainties.

99

5.1 Event generation

The simulated rax distributions of W —► eis and W —> nv events require a large

number of events in the high mx region to minimise the statistical uncertainty.

To facilitate this, a weighted event generator is used and events are generated

with a flat y/1 distribution in the high mx region and reweighted to the Breit-

Wigner distribution using equation 1.24. A leading-order ‘Born level’ Monte Carlo

(MC) event generator is used to generate W —> /m, W —> ev, Z —> ee and Z —>////

events with zero boson px using the CTEQ6M [45] PDFs according to equations 1.22

and 1.24. The scale is taken as the centre-of-mass energy (Q = \ /I ) . The Berends

and Kleiss (BK) [46, 47] algorithm is used to calculate the affect of a radiated

photon, with the Feynman diagrams shown figure 1.3 for W bosons, and a virtual

photon, with the Feynman diagrams shown in figure 1.4.

5.1.1 Electroweak corrections and uncertainties

The uncertainty on Tw from higher order contributions to final state QED radi­

ation, not included in the BK calculation, is estimated by comparing the value

of obtained by using the PHOTOS [48, 49] event generator in the single photon

mode and the two photon mode. In addition to the direct effect on Tw from gen­

erating an additional photon, there is an indirect effect on the m u distributions

in Z boson events from the subsequent change in the values for the calorimeter

response (S'06111) and resolution («), described in section 6.4, and the tracking re­

sponse (S'™0111) and resolution (S res) described in section 6.3. The change in Tw

from the new response and resolution values are added to the change in Tw when

using PHOTOS with an additional photon (denoted ly —> 2q), shown in table 5.1

100

for W —> 6is events and table 5.1 for W —► jiv events.

2 7 5 cem n totalm fit 17

QED radiation.

80 -1 7 +9 - 7 1585 -1 5 + 8 - 6 1390 - 9 +5 - 4 8

1 0 0 - 0 .2 + 2 - 1 1

1 1 0 - 2 + 1 0 1

c uncertainties for W —►eu events, in 1

rarp I 7 —> 2 7 gmom gres total80 - 1 0 +28 -1 4 485 - 1 0 +26 -1 3 390 - 7 +16 - 1 0 1

1 0 0 - 3 +7.6 -5 .4 1

1 1 0 - 1 +4.8 -2 .4 1

Table 5.2: Systematic uncertainties for W —> fiis events, in Me V, from final state QED radiation.

In addition to final state QED radiation, higher-order £non-resonant’ electroweak

corrections to the LO cross-section, dominated by the ‘box’ diagrams shown in

figure 5.1, have an effect on the m ^ distribution, particularly in the high mx

region [50].

z°

d. W~

U , H VuT7[

z°?/,

U, XI l ~ , l +

W~ vi

u , n

d , d

,W~

/

'Z°

\Z°\

.4-

i+,i-

Vi, vi

Figure 5.1: Feynman diagram for non-resonant ‘box’ electroweak contributions to leading order W boson production.

101

The effect of non-resonant electroweak correction on the m t tail region is esti­

mated using the WGRAD event generator [50]. A simplified detector simulation is

used, and is parameterised using a linear response and Gaussian resolution,

obtained from fits to the standard simulation. The fractional change S ewk in the

WGRAD mT distribution from including non-resonant correction is

<ewk _ 0.9986 for mT < 90 GeV

1.020 — 2.2 x 10 4 x mT for mT > 90 GeV

for W —► ei/ events, shown in figure 5.2 (left), and

(5.1)

Sewk 1.0026 — 3.9 x 10 5 x mT for mT < 95 GeV

1.025 - 2.7 x 10-3 x mT for mT > 95 GeV(5.2)

for W —> fiis events, shown in figure 5.2 (right).

V)

0.99

0.98

0.97100 150 200

m,. (GeV)

!w

0.99

0.98

0.97

50 100 150 200nV (GeV)

Figure 5.2: Fractional change S ewk in the mT distribution when including non­resonant contributions using WGRAD for (left) W —> ev and (right) \xv events.

The shift in the measurement of T\y, shown in table 5.3, is added to the fitted

value of T\v presented in chapter 8. The associated uncertainty is estimated as the

change in the value of Tw from varying the Gaussian smearing by an amount

102

equivalent to the uncertainty on the resolution of the recoil model described in

section 6.6.

I I l / r j i b U I l C b l

"80 8~85 1090 11100 15110 20

W —> eis W —> /xz/correction uncertainty correction uncertainty

256 6 6

1112121723

256 6 6

Table 5.3: The correction to Tw and the associated systematic uncertainty, in MeV, from non-resonant electroweak corrections.

The combined systematic uncertainty on Tw from higher order electroweak cor­

rections, shown in table 8.1, is obtained by adding the QED and the non-resonant

systematic uncertainties in quadrature.

5.1.2 PD F uncertainty

The CTEQ6M PDFs are defined by 20 orthogonal parameters with values and as­

sociated errors obtained by minimising the x 2 of a fit to experimental data. The

change in Tw from individually varying each of the parameters by their asym­

metric uncertainty corresponding to a 90% confidence level, is obtained by fitting

the simulated ttt-t distribution to pseudo-data simulated using the default values.

Figure 5.3 shows the values of Tw for each of the PDF parameters for = 90

The combined uncertainty on Tw from the PDF parameter uncertainty is esti­

mated using

GeV.

all parameters(5.3)

103

X 2.11

2.1

2.09

2.08

PDF parameter

Figure 5.3: Variation on Tw obtained by individually varying the CTEQ6M PDF parameters by their associated uncertainty for = 90 Ge V.

where is the variation on Tw from varying PDF parameter i by its 90%

confidence level uncertainty. The result is then divided by 1.6 to give the la

variations, shown in table 5.4.

W —> ei/ W —> /iz/80 15 1685 17 1890 16 16100 19 21110 24 25

Table 5.4: Systematic uncertainties, in MeV, from varying the CTEQ6M PDF pa­rameters.

The CTEQ6M PDFs used in this analysis are calculated to NLO and are chosen

since they provide a more conservative estimate of the uncertainty from the PDF

parameters than the MRST 2004 PDFs [17]. As the latter provide PDFs calculated

to both NLO and NNLO, the uncertainty on Tw from higher order QCD calcula­

tions is found by comparing the values of Tw obtained using the MRST 2004 NLO

and NNLO PDFs, shown in table 5.5 for W —> ev and W —> fiis events.

The combined systematic uncertainty on Tw from the PDFs is shown in table 8.1

104

Table 5.5: Higher-order QCD systematic uncertainties, in MeV, obtained from comparing the NLO and NNLO MRST PDFs.

and obtained by adding the two uncertainties in quadrature.

5.1.3 W boson mass uncertainty

The uncertainty on Tw from the W boson mass value used in the event genera­

tor, shown in table 5.6, is found by varying the mass value by its experimental

uncertainty of 25 MeV [51] and finding the corresponding shift in the measured

value of Tw-

80 18~85 1790 9100 4110 2

Table 5.6: Systematic uncertainties, in MeV, from the uncertainty on the W mass.

5.2 Boson transverse momentum

Higher-order production processes result in a non-zero boson px, as the pT of

additional partons emitted from the initial states must be balanced. This happens

in gluon radiation from the interacting quarks, shown in figure 1.9, and when one

or both of the interacting partons are gluons, shown in figure 1.7. The QCD

calculation describing the boson pT distribution diverges at low energies and a

resummation is performed in the divergent region. The Collins-Soper-Sterman

(CSS) [52] resummation calculation is used to describe the divergent region, and

its transition to the perturbative region at higher pt values. The Brock-Landry-

Nadolsky-Yuan (BLNY) [53] parameterisation of the non-perturbative functional

form in the CSS resummation is used.

The Z boson pt (p |) distribution can be measured to relatively high precision

compared to that of W bosons (p^ ), since both decay products are fully recon­

structed, and can be used to constrain the parameters of the non-perturbative

functional form. As W and Z boson production properties are very similar, the

ratio can be calculated to NLO and is largely insensitive to the non-

perturbative functional form and the associated parameter values [54]. The pa­

rameters obtained from the p% distribution can therefore be used to simulate the

p^f distribution with only a small additional uncertainty incurred.

5.2.1 Z boson transverse momentum

In the CSS resummation, the p^ differential cross-section is obtained from a

Fourier integral of the ‘form factor’ Wqq(b, Q, xi, X2 ) over impact parameter 6,

with the scale Q = 1/b taken as the boson pT. When b > 1 GeV-1 the perturba­

tive calculation of Wqq diverges so Wqq is replaced with

Wqq (b) = W ^ ( b , ) W " p (b) (5.4)

106

where and (b) are the perturbative and non-perturbative form

factors respectively. To ensure IVq|rt is not evaluated in the divergent region, 6*

is defined as

b, = b = (5.5)x/1 + (b/bmax)2

so that 6* « b below 6max and 6* « 6max above. The value for 6max determines the

transition from the perturbative to the non-perturbative part and is chosen here

to be 0.5 GeV-1 for consistency with other studies [53].

The BLNY form for is a Gaussian distribution defined as

= exp ( -b 2 [gx + g2 ln(Q /2Q0) + pip3 ln(100xix2)]) (5.6)

where Qo = 1.6 GeV and the BLNY parameters pi, g2 and p3 are determined from

experimental data. The value for g2 is obtained by minimising the x 2 of the fits

to the P t distribution of Z —» ee and Z —> /i/i candidate events. The parameters

pi and p3 are not well constrained by data taken at a single s value, such as the

Z boson data. Instead, the parameter values obtained from low energy Drell-Yan

data taken over a range of s values are used [53].

The simulated p^ distribution used to obtain g2 includes the effect of the detector

resolution and acceptance, described in chapter 6. As a single fit is made to data

with an average rapidity of |Y| = 0.3, the p \ dependence on rapidity is simulated

using a resummed NLO calculation of ^p (Y = 0 .3 )/^ ^ [55] using the CSS

formalism. The contribution of the processes qq Zg with an additional real or

virtual gluon, and gq —> Zq are included in the perturbative calculation.

To obtain g2, a parabola is fitted about the minimum values of the x 2 obtained by

varying the g2 parameter. The best fit values for g2, with fits shown in figure 5.4,

107

are 0.62 ± 0.08 GeV2 with x 2 — 50.8/48 for Z —>• ee and 0.68 ± 0.05 GeV2

with x 2 = 51.4/48 for Z —> fifx. Both values are consistent with a single model

describing the true p \ distribution for both decay channels. A combined fit by

minimising the x 2 from both decay channels gives g2 = 0.66± 0.04 GeV2 with

X2 = 102.9/96 and is used in this analysis. This value is consistent with the

BLNY value of g2 — 0.68io!o2> obtained using the CTEQ3M PDFs.

O 600

r 500

300

200

100

p? (GeV)

Figure 5.4: The p \ distributions for Z —> ee (left) and Z —» pp, (right) candidate events, fitted to obtain g2 for the W ^ p form factor used in the boson pT parame- terisation.

The parameter values for the BLNY parameterisation of IVNP used in this analysis

are shown in table 5.7.

~gi 0.21 ± 0.01 GeV2g2 0.66 ± 0.04 GeV2g3 -0 .06 ± 0.05 bmax 0.5___________ GeV"1

Table 5.7: Parameter values for the W ^ p form factor used in the boson p r pa­rameterisation.

108

5.2.2 W boson transverse m om entum

To simulate the W boson pt , a resummed NLO calculation of d^ df^ PT / ddYdpT

(which is part of the same calculation used for the p% rapidity dependence men­

tioned above) is used to model the p™ dependence on rapidity and s. The inclu­

sion of the s dependence, shown in figure 5.5 for on-mass and highly virtual W

bosons, is necessary because Z bosons in the fit are produced near the Z pole, by

requiring 80 < m u < 100 GeV, whereas candidate W bosons are selected in the

region 50 < m t < 200 GeV. Also the p™ cross-section dependence on s directly

affects the W boson m t distribution.

0.08

0.06

0.04

0.02

20 40PT (GeV)

r! WFigure 5.5: for on mass-shell and highly virtual W bosons.

5.2.3 Boson transverse m om entum uncertainty

When using the resummed CSS formalism with the BLNY parameterisation con­

strained by Z boson candidate events to simulate the py distribution, the es­

timate of the associated uncertainty has a number of components. These can

be grouped into three categories; the uncertainty associated with the global fit

values for the BLNY parameters which dominate the differential cross-section at

109

low pj!, the uncertainty in the differential cross-section lineshape over the full p™

range, and the uncertainty associated with the calculation of ^ ; ( Y — 0-3) / fy^pT

an(i rf3 w / d2(jZ m iu dY ds dpT ' dYdpT ’

Any skewness in the ^ differential cross-section introduced from the CSS resum-

mation and BLNY parameterisation, in particular from the transition between

the non-perturbative to the perturbative region, is estimated by assuming an ad­

ditional linear pt dependent ‘skew’ parameter (B). The true cross-section can

then be expressed as

d a z ( d a z \— = — X l + B x p i 5.7d p r \ d p x ) b l n y

where ( ^ ^ ) b l n y is the differential cross-section calculated using the CSS resum-

mation with the BLNY parameterisation.

The uncertainty on Tw from the BLNY parameter g2 and the skew parameter

B are estimated by simultaneously varying g2 and B to minimise the y 2 of the

fits to the pz distribution of Z —> ee and Z —> /i/i candidate events. The values

g2 = 0.64 GeV2 and B = —0.0014 GeV-1 are obtained with the covariance m atrix

( 0.0442 -1 .38 x 10-7 ^

. -1 .38 x 10“7 0.0012 ,

which has the corresponding 68% and 95% confidence level contours shown in

figure 5.6 together with the BLNY global fit value. The value for g2 is compatible

with the combined fit values used in the simulation, and B is consistent with zero.

The distribution of the Tw values, obtained by repeatedly sampling the covariance

matrix to obtain sets of g2 and B , are fitted with a Gaussian function and the

110

width is taken as the associated uncertainty on Tw, shown in table 5.8.

p % 0.002 □ blny

— 68% CL— 95% CL

- 0.002

- 0.004

-O.OOg 0.5 0.6 0.7 0.8 g2 (GeV2)

0.9

Figure 5.6: The 68% and 95% confidence level contours for the g2-B covariance matrix with the shaded region showing the B L N Y global fit value for g2.

The uncertainty on Tw from the associated uncertainty on the gi and gs BLNY

parameters is found by varying the parameter values by their 95% confidence

level uncertainty to give the uncertainty on Tw shown in table 5.8.

The calculation of |£ ( Y = 0.3)/ and / s r f e has a negUSible theoret-

ical uncertainty as the uncertainty associated with the choice of interaction scale

and resummation formalism cancel in the ratio. There is a weak PDF dependence

affecting the value of Tw by less than 1.5 MeV. An additional uncertainty from

the value of a s, used in the PDFs when calculating the cross-section ratios, is

found to be less than 1 MeV by comparing the value of Tw obtained using the

MRS-R1 and MRS-R2 [56] PDFs, whose main difference is the value of a s.

The uncertainty on Tw from the boson pT simulation is shown in table 5.8.

I l l

m . 9i 92, B 93 PDFs, total80 2 7 2 2 885 2 7 2 2 890 2 6 2 2 7100 2 6 2 2 7110 2 5 2 2 6

Table 5.8: Systematic uncertainties, in MeV, from the p™ simulation.

112

Chapter 6

D etector sim ulation

The broadening of the W boson mT distribution that results from the detector

effects, has three main components. These are the resolution of the lepton pT

or measurement, the resolution of the recoil measurement, and the rapidity

distribution of accepted W bosons. This chapter describes the simulation of these

detector effects on the mT distribution, with the generated events described in

chapter 5 as input. The associated uncertainty on the measurement of Tw is also

evaluated.

The effect of the detector response and resolution on generated leptons are simu­

lated using a fast parameterised detector simulation, tuned by fitting simulated Z

boson events to candidate events, augmented with input from a full GEANT-3 [57]

simulation of the detector geometry and material. The detector lepton acceptance

is simulated using a combination of simulated detector geometry and a parame­

terisation of acceptance efficiencies obtained from the data. The recoil response

and resolution is simulated using a fast parameterised simulation, also tuned by

fitting simulated Z boson events to candidate events.

113

6.1 z vertex simulation

The primary interaction point varies between events, and the distribution of the

longitudinal component of the interaction point, the z vertex, can be derived from

the beam luminosity function (C) in the z direction. This is proportional to

— zz 2

dC e 2dz 47T(7x(z)(Jy(z)

where ax and oy are the transverse beam widths. The beam widths are propor­

tional to

/?x,yM = /3;,,[i + ( z ~ ! 0x-0y)2] (6 .2 )Px,y

and the z vertex distribution can be expressed as

-*2 g

(6.3)^ [1 + ( ^ ) 2][1 + ( ^ ) 21

assuming (3* = (3* = (3*.

By fitting to minimum bias events over the period data were collected for this

analysis, the values az = 40 ± 0.08 cm, Zqx = 3.00 ± 1.87 cm, zQy = 3.36 ±

1.87 cm and (3* = 43.19 ± 0.17 cm are obtained. The z vertex distribution,

shown in figure 6.1 (left), is simulated by randomly sampling the distribution in

equation 6.3, and is shown in figure 6.1 (right) for W —> ev candidate events.

114

,2 3000

2000

1000

50■50 0

0.8

0.6

0.4

0.2

0 50-50z0 (cm) Zo (cm)

Figure 6.1: The z vertex (left) functional form and (right) distribution forW —> ev candidate and simulated events.

6.2 Silicon tracker sim ulation

Leptons and final state QED photons from the hard collision traverse the sili­

con tracker before reaching the COT, interacting with the detector material. A

description of the silicon tracker is provided by SiliMap, a lightweight param­

eterisation of the silicon tracker material based on the full GEANT-3 simulation

of the CDF geometry and material (CdfSim), tuned to test-beam and collision

data [58]. SiliM ap is a 3 dimensional parameterisation of the geometry and

properties of all the material up to the COT inner layer. SiliM ap contains 32

radial layers, 999 layers in the z direction and a minimum of 120 layers in the <f>

direction, with the exact number depending on the level of detail required and

the distance from the interaction point. The information necessary to simulate

particle interactions with the material, described in section 2.1, is obtained for

each cell using GEANT-3. Leptons, and final state QED photons from the event

generator, are propagated through SiliM ap where interactions with the detector

material are simulated.

115

Bremsstrahlung radiation and pair-production produce secondary particles, which

themselves propagate through the detector interacting with the detector material.

Although the interaction of the secondary particles with the detector medium was

also simulated, it was found that the effect on the measurement of Tw is negligible,

and was therefore not included in the measurement of Tw-

Bremsstrahlung radiation is the dominant energy loss mechanism for electrons

in the silicon tracker and is simulated using equation 2.3, with ymax = 1.0 and

2/min — 0.001. The fractional radiation length is provided by SiliM ap. The

converted photons are distributed in y [20] using

where y is the electron energy fraction converted into a photon.

Pair production is the dominant energy loss mechanism for photons in the silicon

tracker and is simulated using equation 2.4 with the fractional radiation length

provided by SiliMap. The converted photons are distributed in x [20] using

where x is the photon energy fraction converted into an electron or positron.

Ionisation is the dominant energy loss mechanism for muons, and is simulated for

electrons and muons using equation 2.1 with Sternheimer’s parameterisation of 6

(6.4)

(6.5)

for silicon [20]. Values for K and I are provided by SiliM ap.

116

6.2.1 Silicon tracker m aterial scale

A multiplicative scale factor 5 mat is applied to the fractional radiation lengths in

SiliMap, and is found by fitting to the E /p distribution in W —> eis candidate

events in the region 0.8 < E /p < 2.0. The E /p requirement for candidate

electrons is removed so the E /p tail can be included in the fit, since it is sensitive

to the amount of bremsstrahlung in the inner tracker. The resultant increase

in QCD background, described in section 7.3, is reduced by increasing the

requirement to 30 GeV, giving a background of (1.29 ± 0.25)% in this sample.

The fitted E /p distribution with background events included in the simulation,

shown in figure 6.2, gives

Smat - 1.033 ± 0.007stat ± 0.007background (6.6)

H 03„ . 10 0 0 rizzow

800c0>UJ600

400

200

1 1.5 2E/p

Figure 6.2: The E /p distribution of W —> eis candidate events fitted to obtain the SiliM ap material scale factor, with background events added to the simulation.

117

6.2.2 Silicon tracker sim ulation uncertainty

The uncertainty on Tw in W -> e^ events, from simulating only two iterations of

particle interaction in the silicon tracker, is estimated as the maximum change in

the value of Tw when simulating three and four iterations.

Bremsstrahlung radiation is suppressed below ym-m = E /E m due to the Migdal

effect [59], where E is the incident electron energy and E m = 72 TeV for silicon.

For electrons in the region of 25 < E T < 100 GeV, this corresponds to ym[n in

the region 0.0004 < ymin < 0.0014. The uncertainty on Tw in W —► ev events

from simulating bremsstrahlung with ymin = 0.001 is estimated as the maximum

change in the value of Tw using ym[n values in the range 0.005 < ymin < 0.002.

Compton scattering has a non-negligible cross-section for low energy photons,

accounting for around 10% of the total cross-section for 100 MeV photons, rising

to around 60% for 10 MeV photons. To estimate the systematic uncertainty

from not including Compton scattering in the simulation, the latter is simulated

in addition to pair production, according to the pair-production and Compton

scattering cross-sections in silicon [60]. The Compton differential cross-section

(da/dy oc 1 / y + y) is obtained from the Klein-Nishina formula [61]. For W —> ev

events, the change in Tw from including Compton scattering is taken as the

associated systematic uncertainty on Tw-

The uncertainties on Tw from the energy-loss simulation are shown in table 6.1

for W —*■ qv events, and are negligible for W —► fiv events.

iterations bremsstrahlung Compton to tal all 8 8 7 13~~

Table 6.1: Systematic uncertainties for W —> eis events, in MeV, from the energy- loss simulation.

118

The systematic uncertainty on Tw from the material scale, shown in table 6.2, is

obtained from the change in Tw from varying S mat by the quoted uncertainty.

m t“ 80 3”

85 390 2100 1 110 0

Table 6.2: Systematic uncertainties for W —► eis events, in MeV, from the silicon material scale.

6.3 Central outer tracker simulation

The Pt of electrons and muons is determined from the track curvature in the

COT. The measured Pt fluctuates about the true Pt from the quality of the

reconstructed track and local variations in the magnetic field. Any difference in

the overall magnetic field strength results in a pt measurement that differs from

the true Pt by a multiplicative scale factor, defined as the response.

6.3.1 Central outer tracker scale and resolution

The track curvature resolution is defined as

^ P ( )true ( ) measured (^-7)P t P t

where q is the lepton charge. The track curvature resolution is dependent on the

length of the reconstructed track and subsequently on the number of hits. To ac­

count for this, the curvature resolution is found for the different hit combinations

119

using Cdf Sim. The COT response, and an additional global curvature resolution,

are found using Z —> n(i candidate events.

The Ap distributions for the four possible TV8*1211 and N stereo combinations, which

meet the requirements listed in section 4.2, are obtained from CdfSim W p,v

events, shown in figure 6.3 fitted with a Gaussian distribution.

CM

10

1- 0.01 0.005-0.005 0 0.01

o IQ-

10

- 0.01 -0.005 0 0.005 0.01Ap (GeV'1) Ap (GeV1)

o

- 0.01 -0.005 0 0.005 0.01

>a>(37Oy—

<0e4>>111

- 0.01 -0.005 0 0.005 0.01Ap (GeV'1) Ap (GeV1)

Figure 6.3: The Ap distributions of CdfSim W —> pv events for muon tracks with (top) A ^ ial = 4, (bottom) TV8*181 = 3, (left) N stereo= 4 and (right) N stereo= 3.

The CdfSim Ap distributions, with N 8X18,1 and N stereo distributed according to

Z —► /i/i candidate events, are sampled to simulate the pt resolution. The sampled

value of Ap is multiplied by S res to account for a global difference in the pT

resolution between the data and CdfSim. The resultant pt value is multiplied

120

by S mom, the COT response. The values of S res = 1.100 ± 0.039 and S'01001 =

0.9989±0.0004 are obtained by minimising the yfi of fits to the distribution of

Z —> fifi candidate events, shown in figure 6.4. The linearity of the pr response is

confirmed to within 1 a of the quoted uncertainty, since a combined measurement

of the response using J/4t —*■ fi/i and T —> p\i events, which have an invariant

mass of 3.1 GeV and 9.5 GeV respectively, gives S res = 0.9985 ± 0 .0 0 0 2 [61].

O 400

« 3 0 0 -

200

100

90

Figure 6.4: The m MA1 distribution of Z —> /qw candidate events fitted to obtain the COT scale and global curvature resolution.

The angular resolution of the track is simulated by smearing (ft and cot(#) with

Gaussian distributions of widths = 0.002 and crcot(0) = 0.011 respectively, ob­

tained by fitting to the (fttrue - (ftmeasured and cot(0true) - cot(0measured) distributions

of CdfSim W —> pv events.

6.3.2 Central outer tracker sim ulation uncertainty

The uncertainties on Tw for W —► pay events from S res and S mom are estimated

by varying the values individually by —4a,—2a, +2a and +4cr from their nominal

values. The la uncertainty is obtained by interpolation.

121

The uncertainty associated with the CdfSim Ap distribution is obtained by scaling

the ‘central’ region of the A p distribution with respect to the ‘ta il’ regions, both

defined below, and fitting the E /p distribution of W —> eis candidate events to

obtain the scale factor. The central region of the A p distributions is defined as

|Ap| <\ 0.001, and is fitted v/ith a Gaussian. The tail region is defined as | ^

0.001, and is scaled by 5 Ap, which has a corresponding affect on overall resolution

of S res = 1.14 — 0.04 x 5 Ap, so the latter is scaled accordingly. The value for S Ap

is obtained by fitting to the E /p distribution of W —> ev candidate events, shown

in figure 6.5, with > 35 GeV and E ^ / E em< 0.04 to reduce the number

of electrons with large hadronic leakage in the low E /p region. Large binning is

chosen to decouple the track curvature resolution from the calorimeter resolution.

~ 30000-

20000

10000

0.9 1.2 1.3E/p

Figure 6.5: The E /p distribution of W —> ev candidate events fitted to obtain the scale factor applied to the region |Ap| > 0.001 of the A p distribution.

The fitted value is

S Ap = 1.03 ± 0.28stat ± 0.34^ ± O.Ol^mat ± 0.08background (6-8)

with the uncertainties evaluated for the fit (stat), calorimeter resolution («),

silicon material scale (5mat) and background.

122

The systematic uncertainties on Tw from the track curvature resolution distribu­

tions are obtained by varying 5 Ap, which is consistent with unity, by its associated

uncertainty and taking the change in the value of Tw This, together with the

uncertainty from the global resolution and response, is shown in table 6.3.

m fit gmom gves g A p

80 29 29 1785 27 27 1790 17 21 16100 8 11 13110 5 5 10

Table 6.3: Systematic uncertainties for W —> fiv events, in MeV, from the COT scale (Smom), the global COT resolution (Sres) and COT track curvature resolution distributions (SAp).

6.4 Calorimeter sim ulation

After exiting the COT, electrons and photons traverse the Time-of-Flight detector

(ToF) and solenoid, interacting with the material before reaching the calorimeter

towers. Also, constituents of the electromagnetic shower in the CEM may pass

out of the back of the CEM and into the CHA. In addition to the ‘extrinsic’

calorimeter energy loss processes mentioned above, there is an ‘intrinsic’ energy

loss from variations in light yield tha t depend on the depth of the electromagnetic

shower. Both the extrinsic and intrinsic energy loss processes are simulated for

individual electrons and photons, which are assigned to CEM towers providing

they do not traverse the central gap at \z\ < 4.2 cm or the gap between the

wedges at |x| > 23.1 cm from the wedge centre.

The calorimeter resolution parameterises fluctuations in the measured tower en­

ergy about the true constituent electron and photon energy sum, and is simulated

123

together with the response, which parameterises the intrinsic energy loss. Lastly,

the electron E t and Z^had/^em measurement, and the muon Eem measurement

are simulated.

6.4.1 ToF, solenoid and leakage energy loss sim ulation

The total electron and photon energy loss in the ToF and solenoid, and the

energy leakage out the back of the CEM into the CHA, is parameterised using

CdfSim. This takes into account any correlation between the energy loss in the

ToF and solenoid, and energy leakage into the CHA. The detected energy fraction

of incident electrons and photons as a function of their energy, shown in figure 6.6,

is sampled in the simulation.

Any deficiencies associated with the parameterisation of the extrinsic calorimeter

energy loss are absorbed into the non-linear calorimeter resolution parameterisa­

tion, described below, and the Fhad/Eem electron selection efficiency parameteri­

sation described in section 6.5.3.

6.4.2 Calorimeter response non-linearity

The intrinsic calorimeter energy loss has a dependence on ET, resulting in a

different response to incident electrons and photons of different energies. The

reconstructed electron Et is simulated by adding the E T of the constituent elec­

trons and photons (F ^7) after scaling by a + 6 x E ^1 to account for the non-linear

response. The tower Et ,

(6.9)

124

coo2 0.8ZUio

0.4

0.2

102101electron energy (GeV)

10‘1 1 10 102 electron energy (GeV)

coo2ZUJo

0.8

0.6 -

0.4

0.2

102101photon energy (GeV)

photon energy (GeV)

Figure 6.6: The average energy fraction (top) deposited in the CEM as a function of incident energy and (bottom) the distribution of energy fractions, sampled in the simulation for each (left) electron and (right) photon exiting the COT.

can be expressed as

E r = E't '1 + (6.10)

showing the decoupling of the CEM response into a linear and a quadratic (non­

linear) component. The latter is determined from fitting to the data and is

included in the simulation. The linear response parameter (a) is measured after

the inclusion of the non-linear part and is described in section 6.4.3.

Since the pT response is linear, see section 6.3.1, the non-linear CEM response

(b) is determined by fitting to the E /p distribution. The fit is performed in the

125

region 0.9 < E /p < 1.1 to minimise the effect from the material scale. In W —> ev

events, the background is reduced to a negligible level by requiring $ t > 30 GeV.

Simultaneous fits are made to W —> ev candidate events in the region 25 < ET <

50 GeV in 2.5 GeV ranges, shown in figure 6.7, and to Z —> ee candidate events

in the range 30 < ET < 55 GeV. The fits have a x 2/ n<3f of 180/193 and give

the value of b — (267 ± 50) x 10-6 GeV. By fitting to W —> eis candidate events

in 2.5 GeV ranges of E t , the affect of the linear part of the response is largely

decoupled from the non-linear part.

The uncertainty on Tw from the calorimeter response non-linearity, shown in

table 6.4, is obtained by varying the non-linear part b by its uncertainty to give

the subsequent change in the value of Tw-

80 12~85 1390 12100 10 110 6

Table 6.4: Systematic uncertainties for W —> eu events, in MeV, from thecalorimeter response non-linearity.

6.4.3 Calorimeter response and resolution

The measured CEM tower energy is simulated by smearing the true tower energy,

after correcting for a non-linear response, and scaling by a linear response (S cem).

While the CEM resolution quoted in section 2.5.1 is sufficient for most analyses,

this analysis requires a more sophisticated treatment. The constant term (k)

is from variations in response, and can be split into a correlated (kcoit) and an

uncorrelated (Acuncorr) component to describe variations in response over time and

126

a 6oo

400

200

27.5 < E, < 30 GeV

1.050.95E/p

82 300aC?Ul

200

100

25 < ET < 27.5 GeV

1.05E/p

g 2000

1500

1000

50032.5 < Et < 35 GeV

1.05 1.1E/p

0.95

| 1000

500

30 < E, < 32.5 GeV

1.050.95E/p

£ 2000

1000

37.5 < Ef < 40 GeV

0.95 1.05E/p

g 2000

1000

35 < E, < 37.5 GeV

1.050.95E/p

8©I 600?ui

400

20042.5 < Ej. < 45 GeV

0.95 1.05E/p

8og 1500ceiu

1000

50040 < Et < 42.5 GeV

0.95 1.05E/p

100

0.95 1.05

47.5 < Ej < 50 GeV

8o| 200>in

100

45 < E, < 47.5 GeV

0.95 1.05E/p

Figure 6.7: The E /p distributions of W —» eu candidate events in 2.5 GeV ranges of electron E t , fitted to obtain the calorimeter response non-linearity.

127

across towers respectively. The resolution is parameterised as

<t e = 13^5% ^ KUncorr ^ KCOrr ^ 1 ^

E yf Et

with kcovv and / uncorr obtained from data.

Figure 6.8 shows the variation in the mean E /p (< E /p > ) over time, which is

from residual changes in the response after the offline electron reconstruction.

The root-mean-square deviation from the mean value is taken as the correlated

contribution to the resolution, as all electrons in an event are equally affected,

giving Kcorr = 0.29%. The correlated contribution to the resolution is simulated

by randomly sampling a normalised Gaussian distribution for each event, with

width <r = ttcorr, to obtain the multiplicative change in energy, and is applied to

all electrons in the event.

Si 1.065

1.060

1.055

0 50000 100000Sequential even ts

Figure 6.8: The <E/p> distribution as a function of time m W —> en candidate events used to obtain the correlated calorimeter resolution.

The uncorrelated contribution to the resolution and the associated response cor­

rection, are determined by varying ^uncorr and S cem and independently minimising

the x 2 of a fit to the E /p distribution of W —> eis candidate events, and the mee

distribution of Z —> ee candidate events.

128

The fit to the E /p distribution of W —> eis candidate events, with the background

events described in chapter 7 included, in the region 0.9 < E /p < 1.1 gives

Kuncorr = (qM 7 ± 0 .049stat ± 0.147^** ± 0.056smat)% (6.12)

and

S cem = ! 02356 ± 0.00021stat ± 0.00044track ± 0 .0 0 0 1 7 ^ (6.13)

where, in addition to the statistical uncertainty, the uncertainty has contributions

from the track momentum scale and resolution, described in section 6.3.1, and the

material scale S'mat, described in section 6.2.1. Choosing an alternate fit region

of 0.96 < E /p < 1.1 gives Kuncorr = (0.64 ± 0.11stat)%.

4000

£ 3000

2000

1000

0.9 1.2E/p

Figure 6.9: The E /p distribution in W —> eis candidate events fitted to obtain the uncorrelated calorimeter resolution.

The fit to the m ee distribution of Z —► ee candidate events for 86 < m ee < 96 GeV,

shown in figure 6.10, gives ^uncorr = (1.49 ±0.29)% and S cem = 1.02439 ±0.00078.

The values for the response and resolution are combined in a weighted average

using a = E (a j/< rf)/£ (l/o f) where a is the weighted average of individual mea­

surements a* with uncertainty cr*. The combined uncertainty (a) is given by

129

i \ (GeV)

Figure 6.10: The m ee distribution of Z —> ee candidate events fitted to obtain the calorimeter response and uncorrelated resolution.

1/a2 = E(1 /a 2). The values obtained for S cem are consistent, and combine to

give Scem - 1.02382 ± 0.00043.

The three values for /+ncorr are combined, but as the individual values differ

by more than their uncertainty, the two extreme central values are used as the

uncertainty bounds, giving «uncorr = l .0 8 ^ 44%•

The uncorrelated and E t dependent contribution to the resolution is simulated by

randomly sampling a normalised Gaussian distribution with a = 13.5%/y/E r ©

ttcorr for eiectron. The simulated electron energies are then scaled by S cem.

6.4.4 Calorimeter response and resolution uncertainty

The uncertainties on Tw from the calorimeter resolution and response to elec­

trons, shown in table 6.5, are obtained by individually varying the values by

—4cr, —2 (7 , +2cr and +4cr from their nominal values. The la uncertainty is ob­

tained by interpolation. As ftuncorr has an asymmetric uncertainty, the largest

change in the value of Tw from the nominal value is taken as the uncertainty.

130

m fit g c e m K

80 29 4685 27 4390 17 31

1 0 0 7 11

1 1 0 4 4

Table 6.5: Systematic uncertainties for W —> eis events, in MeV, from thecalorimeter response (Scem) and resolution (k).

6.4.5 Electron E ? sim ulation

Individual electrons and photons are assigned to towers, and their energy is

summed using equation 6.9, and smeared to simulate the CEM resolution. The

electron clustering algorithm, described in section 2 .8 .2 , is then performed on the

simulated tower E t deposits to determine the two towers used in the electron E t

simulation.

The Et of electron candidates have a ‘pedestal’ from the contribution from the

underlying event energy, described in section 4.3. Since the latter has a depen­

dence on luminosity, shown in figure 4.2 (right), the electron pedestal also has

this dependence. The luminosity dependence is parameterised as the calorimeter

E £ t , described in section 6 .6 .1 , and the average two-tower underlying event en­

ergy as a function of EE t is sampled in W —► ev candidate events, from a region

at the same 77 and orthogonal in 0 to the electron.

The simulated reconstructed electron Et is the two-tower cluster and underlying

event energy sum, and has the direction of the highest Pt electron track. Although

this track is also used to determine the electron pt , it is not necessarily the

generated electron track as a sufficiently energetic photon from bremsstrahlung

in the silicon tracker may convert into an e+e~ pair.

131

6.4.6 Electron Ehad/^em requirement sim ulation

The electron Ehad/^em distribution, which is used to select electron candidates,

is simulated using the two-tower cluster energy and the simulated hadronic en­

ergy. The latter is obtained by simulating the CHA response and resolution of

the energy leakage out of the CEM into the CHA obtained from CdfSim. The

CHA response is taken as 0.8, the mean E /p for the CEM and CHA energy sum

for events containing a single high pt track [37]. The CHA resolution is simu­

lated as Gaussian distributed with a width of a = 20%/v^E, with the stochastic

term obtained from the best fit to the Ehad/^em distribution of Z —> ee candi­

date events, shown in figure 6.11. This parameterisation is augmented and the

associated uncertainties are evaluated in section 6.5.3.

s 1000

500

0.02 0.04 0.06

Figure 6.11: The Ehad/®em distribution of Z —> ee events, fitted to obtain the CHA resolution for energy leakage out of the CEM.

6.4.7 M uon E em requirement sim ulation

The energy deposited by a muon in the CEM (E em) is used to select muon can­

didates. This is simulated by adding the energy deposited in the CEM, taken

132

from a parameterisation of cosmic ray data, to the simulated underlying-event

energy plus any energy from electrons, positrons and photons ending up in the

same tower as the muon.

6.5 Lepton selection simulation

The geometric selection of leptons from the fiducial detector regions affects the

lepton kinematic distributions, and subsequently the mx distribution of W boson

events. Additionally, the efficiency of lepton selection varies between different

detector regions and as a function of the lepton kinematics. These efficiencies are

included in the simulation and the associated uncertainty is estimated.

Selection efficiencies are obtained from a sample of A otai leptons and the subset

of -/Vpass leptons that meet the lepton requirements, using e = Npass/N tot&\ with

the estimated binomial uncertainty given by cre = y /e( 1 — e)/N tota\.

6.5.1 Lepton rj and (f) dependent selection sim ulation

Electron online selection sim ulation

At least one electron candidate in W —► ev and Z —► ee candidate events has

passed the ELECTR0N_CENTRAL_18 trigger path, described in section 2.8.3. The ef­

ficiency of the ELECTR0N_CENTRAL_18 XFT requirement, described in section 2.8.1,

varies with 7) and affects the lepton 77 distribution and subsequently the Tw mea­

surement. The XFT efficiency as a function of 77 is found in W —> eu candidate

events that have additionally passed the WJJOTRACK trigger path, described in

133

section 2.8.3. This trigger has the same electron calorimetry requirements as the

ELECTR0N_CENTRAL_18 trigger path and no track requirements.

The efficiency of the XFT requirement as a function of g is parameterised as

€(v ) = a0 + ale-'l2/2a' + a2e - ’2/2^ (6.14)

with parameter values, shown in table 6.6, obtained by minimising the x 2 of

the fitted distribution shown in figure 6.12. The online selection is simulated by

clq 0.988(i\ —0.100a,2 0.080a i 0.266a 2 0-102

Table 6.6: Parameter values for the X F T trigger efficiency as a function of g.

weighting generated events in g according to the parameterised XFT efficiency.

0.98

0.96

0.94

•1 -0.5 0 0.5 1

Figure 6.12: The fitted X F T efficiency as a function of

134

M uon online selection sim ulation

Muon candidates have passed at least one of the MU0N_CMX18 and MU0N_CMUP18

trigger paths described in section 2.8.4. The efficiency of each trigger path can

be measured in Z —>• /U/x candidate events by selecting events containing two tight

muons, where one is fiducial in the CMX chamber and the other in the CMUP.

The trigger efficiencies are obtained from the sample that has further passed the

other trigger path to the one being measured and the subset that have passed

both. The obtained efficiencies are e(MU0N_CMX18) = 0.971 and e(MU0N_CMUP18) —

0.894 which are applied as an additional weight to generated events.

M uon chamber sim ulation

Muon trajectories are extrapolated to the simulated muon chamber geometry and

rejected if they do not meet the fiducial requirement, described in section 4.2.

The efficiency of a muon leaving a detectable track, or stub, in the muon drift

chamber that matches a COT track, as defined in Table 4.3, is measured in

Z —»/x/x candidate events for the CMX and CMUP subdetectors. For events th a t

have two fiducial muons in the subdetector being measured, each muon with a

stub is selected as a trigger leg. The muon stub reconstruction efficiencies are

obtained from the sample comprising the other muon in the event, the test leg,

and the subset which meet the stub requirement. The obtained efficiencies are

e(CMX) = 0.988 and e(CMUP) = 0.870 which are applied as an additional weight

to generated events.

135

Electron track requirem ent sim ulation

The efficiency of the electron track requirements as a function of track rj is ob­

tained from the sample of events passing the WJIOTRACK trigger path and the

electron calorimeter requirements, and the subset that additionally pass the elec­

tron track requirements, both described in section 4.1. The efficiency distribution

is fitted with equation 6.14, show in figure 6.13 (left), to obtain the parameter

values shown in table 6.7. The parameterised efficiency is applied as an additional

weight to generated events.

~oq 0.957a\ —3.0000*2 2.991o' i 0.393cr2 0.395

Table 6.7: Parameter values for the electron track selection efficiency as a func­tion of track 7].

M uon track requirem ent sim ulation

The muon track selection efficiency is obtained from Z —> (afi candidate events

since, unlike W —► ytv events, relaxing the track hit requirements does not result

in a significant increase in background. The efficiency is obtained using all the

tight muons, described in section 4.2, as ‘reference legs’ and dividing the other

‘test legs’ into a sample that meet all but the track requirements, and the subset

that additionally meet the track requirements. The efficiency is parameterised as

a third order polynomial with parameter values, shown in table 6 .8 , obtained by

minimising the x 2 of the fitted efficiency distribution shown in figure 6.13. The

parameterised efficiency is applied as an additional weight to generated events.

136

0.95

0.9

0.85■1 0.5-0.5 0 1

w

0.97

0.96

0.95

■1 -0.5 0 0.5 1ti n

Figure 6.13: The fitted (left) electron and (right) muon track hit efficiency as a function of track p.

P o 0.907 P i 0.013 p 2 0.047 p 3 -0.047

Table 6 .8 : Parameter values for muon track hit efficiency as a function of p.

Lepton p and <f> distributions

The lepton p and </> distributions are shown in figure 6.14 for W —► eu and

W —> yns candidate events, with error bars representing the statistical uncertainty

only.

The uncertainty on Tw associated with the p and <fi lepton acceptance simulation,

shown in table 6.9, is obtained from the change in Tw when the simulated events

are weighted so that the p and distributions are identical to the data.

137

o(0I 6000>in

4000

2000

•0.5 0.5■1 0 1

o(0i 6000>w

4000

2000

■0.5 11 0 0.5

5600

5400

5200

50000 2 4 6

o eooo

uj 5000

4000

3000

20 4 6

Figure 6.14: The (top) p and (bottom) <f distributions for (left) W —> ev and (right) W —» pv candidate events.

W —> ev W —► j±vrj (p total p (f) total

80 2 1 3 2 3 485 2 1 3 2 3 490 2 1 3 2 3 4

1 0 0 2 1 3 3 3 51 1 0 2 1 3 3 3 5

Table 6.9: Systematic uncertainties, in MeV, from the lepton p and <f acceptance simulation.

6.5.2 Lepton u\\ dependent selection sim ulation

The efficiency of the lepton selection decreases as proximate event activity in­

creases. It is highest when the recoil component parallel to the lepton (ny) is

138

zero, and lowest when the recoil is in the same direction of the lepton (w|| = \u\).

The efficiency is parameterised as a linear function of u\\ as

:(“ n) =1 + A x u\\ for u\\ < 0

1 + B x u\\ for u\\ > 0(6.15)

with e(w|| = 0 ) = 1 .

The parameters A and B , shown in table 6.10, are obtained from W boson events

in CdfSim, shown in figure 6.15, and also from Z boson events in both CdfSim

and data.

> 0.97

0.965

0.96

0.955

-20 -10 0 10 20

oo- 0.87

0.86

0.85

-20 -10 0 10 20u„ (GeV) u.. (GeV)

Figure 6.15: Lepton selection efficiency as a function of u\\ for CdfSim (left) W —» eis and (right) W —» pv events.

The values obtained from CdfSim W boson events are applied as an additional

weight to generated events. The values obtained from Z boson events are used

to confirm that CdfSim is consistent with the data. The statistical limitation of

this check, estimated as the interval between the W boson value and the furthest

bound of the combined Z boson values, is taken as the systematic uncertainty on

the W boson parameter values.

139

A (xlO ~4) B (x lO -4)z ->ee data 7.1 ± 7 . 8 - 5 . 6 ± 8 . 6z ->ee CdfSim 5.5 ± 1.6 - 8 . 6 ± 1.8w -+ eu CdfSim 3.4 ± 1.0stat ± 3 .7 sys - 8 . 7 ± 4 .6 stat ± 13 .4sysZ -►PH data 2.6 ± 8 . 0 - 2 6 .0 ± 9 . 3z -►pp CdfSim 1.3 ± 1 . 4 - 1 7 .2 ± 1 . 8w - + fJLV CdfSim 4.8 ± 2 .9stat ± 14.8sys — 16.0 ± 5 .8 stat ± 5 .7 sys

Table 6.10: Parameter values for lepton selection efficiency as a function of u\\.

The uncertainty on Tw, shown in table 6.11, is obtained from the change in Tw

when varying parameters A and B by their combined uncertainty.

W —> e v W —► f i v

80 2 785 2 790 2 6

1 0 0 1 2

1 1 0 0 1

Table 6.11: Systematic uncertainties, in MeV, from e(u||) simulation.

6.5.3 Electron E t dependent selection sim ulation

The Et dependence of the electron selection, excluding the -Ehad/^em require­

ment, is taken from CdfSim W —► eu events, shown in figure 6.16 (left). It is

parameterised as

1C + D x (E t - 42) for ET < 42 GeV(6.16)

C for E t > 42 GeVwith parameter values shown in table 6 .1 2 together with param eter values ob­

tained from Z —> ee events in both CdfSim and data.

Equation 6.16, with the value for D obtained from CdfSim W boson events and

140

without scaling with scaling

'1 0.99

5030 40 60

$ 10CN" 0.99

0.98

0.97

0.96 40 50 60 70 Er (G«V)

Figure 6.16: Electron selection efficiency as a function of Et excluding the F’had/ 'em requirement (left) and for only the E’had/^em requirement (right), with the simulated E ^ / E em efficiencies normalised so the average efficiency is equal for all three distributions.

D xlO - 4 GeV- 1 C Z —> ee data 7.1 ± 5 .6 0.98Z —> ee CdfSim 7.2 ± 0.1 0.9789W —> eis CdfSim 4.6 ± 0.4stat ± 3.6sys 0.9845

Table 6 .1 2 : Parameter values for c( E t ) simulation.

C set to untiy, is applied as an additional weight to generated events. The values

for D obtained from Z boson events are used to confirm that CdfSim is consistent

with the data. The statistical limitation of this check, estimated using the same

method as in section 6.5.2, is taken as the systematic uncertainty on the W boson

parameter values.

The uncertainties on Tw associated with the Et dependent selection efficiency,

excluding the Ehg^/Eem requirement, are shown in table 6.13. They are obtained

from the change in Tw from varying parameter D by its combined uncertainty.

The E’had/F'em requirement is excluded from the parameterisation of the electron

selection efficiency as it is included in the calorimeter simulation, described in

141

section 6.4.6. However, an additional ET dependent scale factor (g had/em) is a p ­

plied to simulated W —*■ ev events to match the simulated E em distribution

to that of candidate events, after applying a normalisation factor tha t preserves

the average efficiency. The E ^ ^ / E ^ distribution of W —> ev candidate events

is shown in figure 6.16 (right), together with the E ^ / E ^ distribution simu­

lated both with and without applying the £'had/em scale factor, and normalised

so the average efficiency is equal in all three distributions. The values of S had em

obtained are

ghad/em _ ^

0.9964 for ET < 39 GeV

1.0 for Bp > 40 GeV (6.17)

0.9964 + 0.0036 x (E^ — 39) otherwise

and are applied as an additional weight to generated events.

The uncertainties on Tw associated with scaling E ^ / E em by the ET dependent

factor 5 had/em, shown in table 6.13, is estimated as the change in Tw simulated

with and without including the S had em scale factor. The E ^ / E ^ distribution

in the region ET > 50 GeV is not constrained by W —> ez/ candidate events,

and no scaling is applied. The associated uncertainty on Tw is estimated as the

shift in Tw obtained when applying a linear scale factor in the ‘T^had/^em ta il’

region of 50 < ET < 100 GeV, determined by matching the simulated E ^ /E e m

distribution to that of CdfSim.

142

ra^ e (£ r) 5'had/em E ^ / E em tail total808590100110

121 2

977

13131 0

Table 6.13: Systematic uncertainties for W —► ei/ events, in MeV, from e(ET) simulation.

6.5.4 M uon p t dependent selection simulation

The muon selection, described in section 4.2, has a pt dependent F^ad threshold

resulting in a p r independent selection efficiency, shown in figure 6.17 (left). The

P t dependent £?had threshold is obtained from the muon (T^ad) distribution as

a function of p r in CdfSim W —> events, which has a linear fit, shown in

figure 6.17 (right), of Ekad = 5.6 + 0.014 x pT GeV.

The systematic uncertainty on Tw is taken as the change in Tw from varying the

slope of the £had requirement by its statistical uncertainty, giving an uncertainty

of 1 MeV.

60 70PT (GeV)

A

111V

8040 60 100PT (GeV)

Figure 6.17: Muon selection efficiency as a function of Pt (left) for W —> /iv candidate events and (right) muon (i?had) as a function of p-r for CdfSim events.

143

6.6 Recoil simulation

The recoil, described in section 4.3, has a ‘hard’ component from QCD radiation

from the boson production, and a ‘soft’ component from the spectator parton

interactions and minimum bias events. The hard recoil component is primarily

the detector response to the radiation balancing the boson pt - Hence the hard

recoil component shows a strong correlation with boson p r, increasing with the

Pt and predominantly directed anti-parallel to it. The soft recoil component

is strongly correlated with luminosity since it is dependent on the rate of pp

interactions.

In order to decouple the hard and soft recoil contributions, the recoil (u) is pa­

rameterised in terms of a parallel component (u\) and a perpendicular component

(1L2 ) with respect to the boson pr- As the latter is not well measured for W boson

candidates, the parameters of the recoil model are determined from Z boson can­

didate events, since pf- is well measured. The recoil simulation is parameterised

in terms of the true boson P t, described in section 5.2.

6.6.1 sim ulation

Both the recoil and the simulated electron E r , described in section 6.4.5, have a

dependence on luminosity that is included in the simulation. As the luminosity

and the calorimeter £ £ r , defined in section 4.3, are highly correlated, shown

in figure 6.18, the latter is simulated as it is an explicit measure of calorimeter

activity.

To include correlations between the boson pT and the E E T distribution, the latter

144

60

50

40

0 20 40 60Luminosity x 1050 (c m 'V )

Figure 6.18: £ £ t as a function of luminosity for W —> en candidate events.

is parameterised as

£ E t = (Qi + Q2 x P t ) x r (Q 3 + Qa x P t ) (6.18)

where T is the gamma function and pp is the simulated true boson p r • The

parameters are obtained by minimising the y 2 sum of simultaneous fits to the

distribution and the mean Y,ET ((E£'T)) as a function of p | , shown in

figure 6.19. The value of the parameters is determined separately for Z —» ee and

Z —> fifi candidate events, and shown in table 6.14.

Z —> ee Z —> pp“ Qi 20.700 19.351

Q2 -0.154 -0.141Q3 2.055 2.051Q4 0.085 0 .1 0 0

Table 6.14: Parameter values for the £ £ t parameterisation.

145

y2/ndf = 10/22

200 250I E,. (GeV)

$ 1000X2/ndf = 25/21

800

> 600

400

200

100 150 200 250I Ej (GeV)

0 5 10 15 20 25 30 35 40 45P? (GeV)

>o> 85 O7 80

7570

^ (GeV)

Figure 6.19: The (top) HE? distribution and (bottom) ( £ £ r ) as a function of Pt , fitted to obtain the EE^ parameters for (left) Z —» ee and (right) Z —> fifi candidate events.

6.6.2 Soft recoil resolution simulation

The direction of the soft recoil components is defined along the x and y axis (ux

and uy respectively). The components are parameterised as Gaussian distributed

quantities with (uXtV) = 0 and with a resolution (cr) parameterised as

<j(ux,„) = /?! x (S £ t ) B2 (6-19)

with parameters R\ and R 2 obtained from minimum bias events with a tower

threshold of 100 MeV. By simultaneously minimising the x 2 of the fits to a(ux y)

146

as a function of £Pt> shown in figure 6 .2 0 , a combined soft recoil resolution (crMB)

with parameters R\ = 0.3384 and R 2 = 0.5589 is obtained.

Figure 6.20: The soft recoil resolution along the (left) x axis and the (right) y axis as a function of E P t in minimum bias events, fitted to obtain the soft recoil parameter values.

6.6.3 Recoil resolution and response sim ulation

The recoil components u\ and U2 are parameterised as Gaussian distributed quan­

tities. Since u2 is primarily sensitive to the soft recoil contribution, which has

a uniform azimuthal distribution, the mean response of U2 ((112)) is zero in the

parameterisation. Conversely, u\ is primarily sensitive to the hard recoil contri­

bution which is strongly correlated with boson P t , and the mean response of U\

((U\ )) is parameterised in terms of the true boson pT as

(Ul) = (Pi + P 2 X p T ) X (1 - e- p3XpT) (6.20)

with the parameters Pi,2,3 determined from fits to Z boson candidate events.

The resolutions (cr) of U\ and U2 are defined as the widths of their Gaussian

147

distributions, and are parameterised in terms of the true boson pt and (Tmb, as

a(ui) = crM B x {Pa + P5 x pT) (6.21)

and

cr(u2) = cfmb x (P6 + P7 x pT) (6.22)

with the parameters 5 ^ ,7 also determined from fits to Z boson candidate events.

The recoil parameters, shown in table 6.15, are obtained separately for Z —> ee

and Z —> /i/i candidate events by minimising the total x 2 of simultaneous fits to

(iti), cr(wi) and <7 (^2) distributions as a function of p \ shown in figure 6.21. The

uncertainties quoted are taken from the diagonal elements of the 7 x 7 covariance

matrix. The total x 2 of the fits is 26/24 for Z —» ee candidate events and 29/24

for Z —> /i/i candidate events.

W —> eis W —> /ii/Pi -12 .7 ± 1 .1 -13.1 ± 1.5P2 -0.589 ± 0 .0 2 -0.586 ± 0 .0 2

Ps 0.043 ± 0.003 0.042 ± 0.004Pa 0.93 ± 0.03 0.97 ± 0 .0 2

P5 0.019 ± 0.003 0.016 ± 0 .0 0 2

Pe 1.025 ± 0.03 1.064 ± 0 .0 2

Pi 0 .0 0 0 2 ± 0 .0 0 2 - 0 .0 0 2 ± 0 .0 0 1

Table 6.15: Parameter values for the recoil response and resolution, with their statistical uncertainties obtained from the fits to the Z data.

Figure 6 .2 2 shows the comparison between simulation and da ta of the distribu­

tions of u , u\ ,U2 and the angle (A<j>) between the boson pT direction and the

recoil for Z boson candidate events.

148

6.6.4 Recoil sim ulation for W boson events

Since the production processes for W and Z bosons are very similar, the recoil

model and parameters, obtained for Z boson events, are used to simulate the

recoil for W boson events. The lepton and neutrino travel in opposite directions

in the boson rest frame, so the boson to t is given by ~ 2 ET + u\\ and

?0)(3

IV

-15-20

-25

-30

(GeV)

S' 0S -5

£ -10

-15-20

-25

-30

-35

5.5

4.5

3.5

O- 6.5

5.5

4.5

3.5

5.53"

4.5

3.5

4.4

3~ 4.2

3.8

3.6

3.4

3.2

0 5 10 15 20 25 30 35 40 45

Figure 6 .2 1 : The {u\), a{u\) and (7 (112) distributions (rows 1-3 respectively) as a function of p^ for (left) Z —► ee and (right) Z —> fifi candidate events, fitted to obtain the recoil parameter values.

149

> 350 «(9 300(A

250c>ill200 r

150

20 25 3(u(GeV)

® 700Om 600€> 500in

400

300

200

100

25 30u(GeV)

20

yvndf = 34/30

il (GeV)

>«O" 500MC

300

200

100

-25 -20 -15 -10u, (GeV)

X2/ndf = 40/28400 rE 350£ 300

250 r

u, (GeV)

1000 . x2/ndf = 39/30

800

600

400

200

-102 (GeV)

CM F“6500- c §|3 400 7

300 r

200 r

100

1400

<3 1200

1000

800

600

400

200

A<)>

Figure 6 .2 2 : The distributions of u, u\, u<i and the angle A (j> between u and p^ (rows 1-4 respectively) for (left) Z —> ee and (right) Z —> pp candidate events.

150

is directly affected by u\\. Consequently, it is essential tha t the simulated u\\

distributions provide a good description of the data. The mean values of ii||,

shown in table 6.16, are consistent between data and simulation.

The u, u\\ and u± distributions are shown in figure 6.23 for W —> ev and

W events with the background contributions, described in chapter 7, added

to the simulation. Additionally, the distribution of (u\\) as a function of m/p, u

and the angle (A<f>) between the recoil direction and the lepton direction, together

with cr(u±) as a function of u, are shown in figure 6.24.

data simulation W —>ez/ -0 .53 ± 0.02 -0.52W -> \l v -0 .48 ± 0.02 -0.48

Table 6.16: Mean u\\, in GeV, for data and simulation.

6.6.5 Recoil sim ulation uncertainty

Although the correlation between the parameters used to simulate u\ (Pi,.. ,5 )

and those used to simulate w2 (^ 6,7) is negligible, there is strong correlation

between the U \ response parameters (Pi,2 ,3 ) and between the individual resolution

parameters (P4 ,5 and Pqj). Consequently, the systematic uncertainty on Tw is

evaluated by sampling the recoil parameter covariance matrix to obtain 250 sets

of recoil parameters. The corresponding distribution of Tw values, shown in

figure 6.25 for = 90 GeV, is obtained by fitting the multiple w t distributions,

simulated using the 250 sets of recoil parameters, to pseudo-data simulated using

the default recoil parameters shown in table 6.17. The width of the best fit

Gaussian distribution is taken as the uncertainty on Tw-

151

ra^ W —> ez W —> fin80 60 5385 59 5190 54 49

1 0 0 38 301 1 0 19 14

Table 6.17: Systematic uncertainties, in MeV, from the recoil simulation.

152

CO

cm0i-sa>

05 Oia- •asSi-

3o>

o a 3

- J 3 ©O) £•

S :asa-

«: &05& 3-3 2 .sx <=- _ s ^ S'.cS- °^ 05

i•Cs05c2O)3

3£05I

305

S ' ©03 ^ + .CO .<5 05O-M53

Events / GeV

S IO

Events / GeV

Events / GeV

_c

O®5

Events / GeV

j:O<ss

Events / 0.5 GeV

cO«5

Events / 0.5 GeV

cO«5

Events / GeV

t*

O»<

Events / GeV

ts

X2/ndf = 21/10

100 15050m, (GeV)

10 20u (GeV)

«o

A

u (GeV)

X2/ndf = 15/10

12010080rrij (GeV)

X2/ndf = 19/20

A_

u (GeV)

A <(>

-i

u (GeV)

Figure 6.24: Comparison of (u||) as a function of m T, u, and A </> (rows 1-3 respectively) and a(u±) as a function of u (row 4) for simulated and candidate (left) W —> eu and (right) W —► fiv events.

154

0 80aCOo® 60w

A r w = 49 MeV

?00Ess

40

IBn. 20

oSoc 2.1 2.2

0Ocoo©A r w = 54 MeV

w12 E2nQ.OOC 2.3

rw (GeV)2.1 2.2

Figure 6.25: Values of Tw obtained from sampling the recoil parameter covari­ance matrix for to obtain the recoil parameter values for (left,) W —> eu and (right) W —> pis events and fitting to pseudo-data simulated using the default recoil pa­rameter values.

155

Chapter 7

Background events sim ulation

Candidate W boson events have significant ‘background’ event contributions from

other electroweak and non-electroweak processes which affect the rriT distribution

and subsequently the measurement of Tw- This chapter describes the determina­

tion of the event fraction and distribution of background events relevant to

this analysis, and the evaluation of their associated systematic uncertainty. The

background event m T distributions, normalised using the background fraction,

are added to the simulated ttlt distribution, described in chapter 6 .

When describing the background event processes, leptons are not explicitly dif­

ferentiated from antileptons. Also, neutrino flavours are not explicitly given and

are implied by the context. When the charged lepton flavour is not indicated,

the electron and muon are implied and, depending on the context, the tau lepton

also.

156

7.1 Electroweak background events

The electroweak background events for W —> eu candidate events are W t v

and Z / 7 * —> t t events, where a r decays to e v v , and Z /7 * —► ee, events where one

of the electrons is not identified. Similarly, the electroweak background events for

W —► fan candidate events are W —> t v and Z /7 * —> t t events, where a r decays

to finn, and Z / 7 * —> fifi, events where one of the muons is not identified. The

background and signal (W —► en and W —> fin) processes are simulated using

CdfSim. The background fraction is obtained from the ratio of background to

signal events passing the W event selection, described in chapter 4.

7.1.1 W t v background

The W —> t z / background event fractions, shown in table 7.1, are obtained using

events simulated with both CdfSim and the ‘standard’ simulation, described in

chapters 5 and 6 . The combined values are used, and the associated uncertainty

on Tw is obtained from the change in Tw when varying the background fraction

by its associated uncertainty. It is found to be negligible in both cases.

W —► QIS W fjLl/CdfSim 2.04 ± 0.01 1.98 ±0.01standard 2.01 1.98combined 2.04 ± 0.03 1.98 ±0.01

Table 7.1: The W t v background event fraction in percent.

The background tut distributions are taken from the standard simulation, shown

in figure 7.1, as they have more events than CdfSim. The systematic uncertainty

from the shape is obtained from the change in Tw between using the raT distri­

157

bution from CdfSim and the standard simulation. It is found to be negligible for

W —> /ii/ events and has the values shown in table 7.2 for W —> eis events.

5awc0)>ill

104

50 100 150 200

%G<0c>Hi

104

100 20050 150m,. (GeV) m,. (GeV)

Figure 7.1: The m? distributions of W —> t v background events in (left) W —» eis and (right) W —> (iu events simulated using the standard simulation.

80, 85, 90 3 100 51 1 0 8_

Table 7.2: Systematic uncertainties, in MeV, from W —> t v background events in W —► eis candidate events.

T.1.2 Z/7* —> ££ background events

The Z /7 * —> i t background event fractions, where the lepton is an electron, muon

or tau lepton, are obtained from the acceptance ratio of Z /7 * —> I t to signal

W tv events, generated in equal numbers, divided by the cross-section ra­

tio R = <j(W —► tv)/<j(Z /7 * —> tt) . Since the cross-section ratio for Z /7 * pro­

duction is the same as that for pure Z production in the region 6 6 < mz/7*

< 116, the numerator of the background fraction only includes events gen­

erated in this region. This allows the use of the NNLO cross-section ratio

158

a(W -> t v ) / ct{7j -> U) = 10.67 ±0.15 [62].

W boson events with an additional reconstructed track are rejected (Z veto events

described in section 4.5). Since CdfSim has a higher tracking efficiency than

candidate events in the region \rj\ > 1, the number of Z veto events excluded

from the signal W boson events is over-estimated. A scale factor of

, 0.166 ±0.013 for rcot < 83 cmSveto = { (7.1)

0.754 ± (0.0017 ± 0.0006) x r cot for 83 < r cot < 132 cm

determined from data is applied to the number of Z veto events.

The Z /7 * —»11 background event fractions are shown in table 7.3, with subscripts

R and Sveto to denote the uncertainty associated with the cross-section ratio and

CdfSim Z veto scale factor respectively.

W ev

ESI

N

* *

1 1-+ ee

-> T T

0.167 ±0.002# ±0.002stat 0.115 ±0.002* ± 0.001stat

± 0.005*veto

W ^ up

Z/7* - TT 5.66 ± 0.08* ± 0.02stat ± O.ISjS'vetoZ /7 * - -+ T T 0.123 ±0.001* ± 0.001stat

Table 7.3: The Z/y* —> I t background event fraction in percent.

The Z/y* —► t l background event rax distributions are obtained from CdfSim

events generated for raz/7* > 20 GeV, shown in figure 7.2. Due to the low

number of events in the high m t region, the rax distributions are parameterised

as a Gaussian function in the low region, except for Z/y* —► pLpL which is

sampled from the distribution, and a Landau function in the high rax region,

159

with the regions defined in table 7.4.

>«ooT-(0c«>UJ

50 100 150 200

<oi

10050 150 200m,. (GeV) (GeV)

«Oo1“0>

50 100 200150

>«Oo 10:r -

(0s111

10

50 100 150 200m,. (GeV) itv (GeV)

Figure 7.2: The fitted m,T distributions of (top left) Z /7 * —> ee? (top right)Z /7 * —> n/i and (bottom) Z /7 * —» t t background events in (left) W —► eis and (right) W —> fiis events.

Gaussian LandauW - ->• e n

Z /7 * —> ee 50 < mT < 95 95 < mT < 2 0 0

Z /7 * —► T T 50 < ra T < 83 83 < mT < 2 0 0

W - ■> /ii/

tsi * - > /i/i none 90 < mT < 2 0 0

Z /7 * —► T T 50 < m T < 85 85 < m T < 2 0 0

Table 7.4: Parameterisation of the Z /7 * —> i t background event ttt-t distribution, with the associated mx regions in Ge V.

The systematic uncertainty from the Z/y* —> t l background event fraction is esti-

160

mated as the change in Tw from varying the background fraction by its combined

uncertainty, and is found to be negligible for all cases except for the Z —» fi(i back­

ground fraction in W —> giv events, which has the values shown in table 7.5. The

systematic uncertainty from the shape is obtained by varying the fit parameters

by their uncertainty, and is found to be negligible in all cases.

80 lfT 85 1790 14100 5110 3

Table 7.5: Systematic uncertainties for W —► (iv events, in MeV, from Z —> fifi background events.

7.2 Decay-in-flight background events

Charged pions and kaons from the hard collision decay into yiv. Due to their

lifetimes, these decays can occur in the COT. Although the pt of muons from

pion and kaon decays are usually much lower than that of muons from W —> pis

events, the ‘kink’ in the track from the decay in the COT can result in a fake

high pT measurement, and subsequently a large measurement. As the pu

branching ratio is large (63.4% for kaons and 99.99% for pions) they contribute

a significant background to W —> fiu events.

Due to the kink in the meson-muon track, the track has a different Xtrack/ n<lf

distribution compared to true W —* pv muons, with a larger average value. The

‘decay-in-flight’ background event fraction is found by fitting the x?rack/ndf distri­

bution of true W —> pv muons added to decay-in-flight muons, to that of W —► pv

161

candidate muons. The amount of background is varied to minimise the x 2 °f the

fit. The background fraction is found separately for tracks with and without

hits in the silicon tracker. The Xtrack/ n^ distribution of true W —> /ir' muons is

taken from Z —> /i/x events, which have a negligible level of background. Since

the decay-in-flight muons tend to have a large do distribution, a sample of decay-

in-flight muons is obtained by removing the Xtrack/ ndf requirement and requiring

0.15 < |do | < 1.25 cm for tracks with hits in the silicon tracker (0.25 < |do | <

1.25 without silicon hits). The upper do limit is required to provide a sample of

decay-in-flight muons with Xtrack /ndf independent of do, shown in figure 7.3, since

the Xtrack/ndf distribution as a function of do is not consistent with flat above

this value.

A5 3.6

Jf 3.4

3.2

0.5 1 1.5

A5c

cJV

3.5

2.5

0.5 1 1.5ld „l ( c m > ld 0 l <c m >

Figure 7.3: The Xtrack/ndf distributions as a function of |do | for decay-in-flight muons for tracks (left) with silicon hits and (right) without. The dashed line delineates the decay-in-flight |do | range and its < X track / n d f > value.

The background fraction fits to the x 2rack/ndf °f W —► nv candidate events, shown

in figure 7.4, gives a background fraction of (0.113±0.047)% for tracks with silicon

hits, and (0.46 ± 0.17)% without silicon hits. Since 6 .2 % of W —> / i v candidate

events do not have tracks with silicon hits, and the other backgrounds comprise

8.045%, the total decay-in-flight background is (0.146 ± 0.049)%.

162

105

0 2 4 6

C / n d f

0 2 4 64^K/ndf

Figure 7.4: 77ie Z —> /i// plus decay-in-flight (DIF) track Xtrack/11 distribution fitted to W —► /iz/ candidate event, for tracks (left) with silicon hits and (right) without, to obtain the background fraction.

The decay-in-flight raT distribution, shown in figure 7.5 (left), is parameterised

as a Landau distribution, and obtained from W —> pv candidate events without

the Axcmu , Axcmp and A x cmx requirements and with 0.25 < |do| < 0.6 cm.

* I(3

o l - o

50 1 00 1 50 200nv (GeV)

200 irij (GeV)

Figure 7.5: The decay-in-flight m t distribution obtained from W —» pv candidate events using (left) 0.25 < \ d0 | < 0.6 cm and (right) 0.6 < \ d0 \ < 1.2 cm.

The systematic uncertainty from the shape is obtained from the change in Tw

when varying the fit parameters by their uncertainty, and also by using a

distribution obtained with 0.6 < |d0| < 1-2 cm, parameterised as a Gaussian

163

function in the region 50 < mT < 75 GeV and a Landau function in the region

75 < raT < 200 GeV, shown in figure 7.5 (right). The systematic uncertainties

on Tw from decay-in-flight background events are shown in table 7.6, where the

uncertainty from the background fraction is found by varying the fraction by its

uncertainty.

m !p fraction fit parameters do range total “ 80 15 8 6 18

85 18 9 10 2290 21 10 13 27100 29 15 23 40110 38_________ 20_________ 37 57

Table 7.6: Systematic uncertainties, in MeV, from decay-in-flight background events in W —> [ i is candidate events.

7.3 QCD background events

Events where the colliding pp’s interact via QCD are abundant at hadron col­

liders. These ‘QCD events’ often contain hadronic radiation in the form one or

more collimated cascades (jets) containing a large number of particles. Events

with a jet traversing an uninstrumented detector region often have a large P t

measurement. The event will pass the W —> ev selection requirements if it has

sufficient and a jet containing a charged pion meeting the electron track re­

quirements, and a neutral pion that decays into photons meeting the calorimeter

requirements. Additionally, kaons and B mesons, from heavy quark radiation,

may decay semi-leptonically with a ev or fiv in the final state th a t passes the W

boson selection requirements. Since QCD events are common, they contribute a

significant background to W —> ev candidate events.

164

The QCD background fraction is estimated by fitting the distribution of QCD

events, added to true W —> Iv and electroweak background events, to that of

W —> tv candidate events, without the {2^ and m t requirements. The amount of

QCD background is varied to minimise the x 2 of the fit? performed in the region

5 < rriT < 60 GeV. The QCD background fraction is obtained from the number

of events remaining after the requirement is applied. The distributions of

true W —► £1/ events and electroweak background events are obtained from the

standard simulation and CdfSim respectively. QCD events are obtained from

data.

7.3.1 W —► f i v QCD background events

Since true W —»juz/ muons have low levels of proximate calorimeter and track

activity, QCD muons are obtained using the muon selection requirements, de­

scribed in section 4.2, without the J had and Eem requirements, and the W boson

selection requirements, described in section 4.5. The track isolation requirement

is reversed to only select muons with other tracks nearby. This ‘anti-isolation’

requirement is fulfilled by muon tracks that fail the track isolation requirement of

^a# < o .4 ^ £ Qey Decay-in-flight background events are reduced by requiring

the muon track to have hits in the silicon tracker.

The distribution of QCD events is corrected for W — > / i v , W —> t i s and

Z —> nn contamination, with their fractions and f a distributions obtained from

CdfSim. The overall electroweak contamination fraction is obtained from the

W [iv acceptance ratio of CdfSim to candidate events, using the standard se­

lection requirements and correcting for the other background events. The

distribution of anti-isolation muons before and after correcting for electroweak

165

and decay-in-flight contamination is shown in figure 7.6 (left) for an anti-isolation

value of 4 GeV (J^A‘R<0'4 P t > 4 GeV).

Since the QCD background in Z —► pp candidate events is negligible, the number

of accepted Z pp events in CdfSim and data, after requiring an anti-isolation

muon, should be equal. The fractional deviation and its associated uncertainty is

taken as the uncertainty on the electroweak background fraction in anti-isolation

events.

The QCD background fraction is obtained by fitting the distribution of sim­

ulated W —> pv events, with QCD, electroweak and decay-in-flight background

events added, to the distribution of W —► pv candidate events.

IUi

10, 40

Oin — m easured

10

I111

40 60

Figure 7.6: The distribution of (left) the anti-isolation sample before and after the subtraction of electroweak and decay-in-flight contamination and (right) W —> pv candidate events fitted with simulated events added to background events.

Since f a and anti-isolation may be correlated in QCD events, the background

fraction is determined for QCD event samples obtained with anti-isolation val­

ues of Y1AR<0AP t > 4, 6 , 8 , 10, 1 2 , 14 and 16 GeV, shown in figure 7.7. The

uncertainty for each anti-isolation value is the combined uncertainty of the fit to

the distribution, and the electroweak contamination uncertainty. The mini­

166

mum of the fitted quadratic function occurs at an anti-isolation value of 4 GeV.

A background fraction of (0.294 ± 0.008fit )% is obtained from the fit to the

distribution using an anti-isolation of 4 GeV, shown in figure 7.6 (right), and is

taken as the QCD background fraction in W —> fiv events.

■oc32

0.8

22 0-6 _ aoo° 0 . 4 -

0.2

0 10 155anti-isolation (GeV)

Figure 7.7: Fitted W —► fiu QCD background fraction obtained for anti-isolation values ofY2AR<0APT > 4> 6, 8, 10, 12, 14 and 16 GeV.

The additional uncertainties on the background fraction are 0.12% from the

electroweak contamination of the anti-isolation sample, 0 .0 0 1 % from the decay-

in-flight background uncertainty, 0.042% from the difference between the back­

ground fraction obtained for an anti-isolation of 4 GeV and 8 GeV, and 0 .0 2 %

from comparing CdfSim with the standard simulation for the distribution of

true W —► (iu events. An additional uncertainty on the background fraction of

0.06%, from removing the > 50 GeV, is estimated as the subsequent frac­

tional increase in the number of W —► fiv candidate events. The uncertainties are

combined to give a QCD background fraction of (0.29 ± 0.13)%.

The QCD m t distribution, shown in figure 7.10 for an anti-isolation of 4 GeV,

is corrected for electroweak and decay-in-flight contamination and parameterised

as a Gaussian function in the region 50 < m t < 90 GeV and a Landau function

167

in the region 90 < mT < 200 GeV.

10

1

200100 150m , (GeV)

Figure 7.8: The fitted QCD m t distribution obtained with an anti-isolation re­quirement of 4 GeV.

The uncertainty on Tw from QCD background events in W —> /iv candidate

events, shown in table 7.7, is estimated as the change in Tw from varying the

background fraction by its uncertainty, and the shape of the mx distribution. The

parameterised shape of the mT distribution is varied by each fit parameters’ un­

certainty, the electroweak contamination uncertainty and the anti-isolation value.

Tot fraction fit parameters anti-isolation EWK total “ 80 3 5 3 3 7

85 3 5 5 3 8

90 4 5 10 2 12100 7 7 13 3 17110 9___________9____________ 16_________ 4 21

Table 7.7: Systematic uncertainties for W —> yns events, in MeV, from QCD background events.

7.3.2 W —> e v QCD background events

Since true W —> ez/ electrons have low levels of proximate hadronic calorime­

ter activity, QCD electrons are obtained using the electron selection require-

168

ments, described in section 4.1, without the E /p requirement and with the

Eh&d/Eem requirement modified to Ehad/Eem < 0.125. To reduce non QCD elec­

tron background events, such as true W —> en events, three ‘anti-electron’ subsets

are obtained using the following anti-electron requirements; E ^ d /E em > 0.07,

| A;?| > 0 .5 cm and Xstrip > 10? where Xstrip *s the comparison of the shape of the

charge distribution in the 1 1 CES cluster strips to test beam data, taking the to­

tal cluster energy into account. The ‘standard’ anti-electron sample is obtained

by requiring two of the three anti-electron requirements. The ‘-Ehad/Eem’ an<

{Xstrip’ anti-electron samples are obtained by demanding only the E ^ ^ / E ^ or

Xstrip anti-electron requirement respectively (the sample obtained by demanding

only the A z anti-electron requirement contains too few events to be meaningful).

The distributions of the anti-electron samples are corrected for W —> ei/,

W — ► T z q Z — > ee and Z — > t t contamination, with their fractions and dis­

tributions obtained from CdfSim. The overall electroweak contamination fraction

is obtained from the W —> ev acceptance ratio of CdfSim and candidate events,

using the standard selection requirements and corrected for other background

events. A further correction is applied to the W —► ev and Z —> ee contamination

fraction, and accounts for the difference in the ratios of anti-electrons to candi­

date electrons. This ratio is obtained from Z —> ee events in CdfSim and data,

which has negligible QCD background, and is shown in table 7.8.

anti-electron i?data ^cdfsim _Rdata cdfsimStandard (0.17 ±0.05)% (0.068 ± 0.009)% 2.50 ± 1.25£had/£em (1.20 ±0.11)% (2.32 ± 0.05)% 0.52 ±0 .05Xstrip________ (4.66 ± 0.28)% (2.88 ± 0.06)% 1.62 ± 0.10

Table 7.8: The ratio of anti-electrons to standard electrons in data (Rd&t&) and CdfSim (EcdisimJ and the ratio of the two (,i2data/i? cdfeim>).

Since the contamination from W t v and Z t t events is predominantly from

169

hadronic r decay, the anti-electron ratio correction is not applied. The W —► t v

and Z —> t t contamination fraction uncertainty is taken as twice the uncertainty

on the anti-electron ratio correction. The distribution of anti-electrons before

and after correcting for electroweak contamination is shown in figure 7.9 (left).

The QCD background fraction is obtained by fitting the distribution of sim­

ulated W —> ez/ events, with QCD and electroweak background events added, to

the distribution of W —> eis candidate events, shown in figure 7.9 (right) using

standard anti-electrons as the QCD electron sample.

— measured— corrected

UJ 102

40 60

>a>ato

« 104ca>ui

0 20 40 60 80

Figure 7.9: The distribution of (left) the standard anti-electron sample before and after the subtraction of the electroweak contamination and (right) W —> eis candidate events fitted with simulated events added to background events.

The QCD background fractions obtained for the three anti-electron samples are

shown in figure 7.9 together with the uncertainty from the fit and the electroweak

contamination.

The uncertainty on the QCD background fraction from the anti-electron defini­

tion is taken as 0 .2 2 %, which is the greatest difference between the two extreme

central values. The additional uncertainties on the QCD background fraction are

0.01% from using CdfSim for the ‘true’ $ T distribution instead of the standard

170

anti-electron background fraction Standard 1.35 =t O.Olfit ± 0.17ewk-F'had/Eem 1.49 ± O.Olfit ± 0.14ewkXstrip________ 1.13 ± O.Olfit ± 0.17ewk

Table 7.9: The QCD electron background event background fraction, in percent, obtained using the different anti-electron definitions.

simulation, and 0.05% from removing the rax requirement of 50 GeV, estimated

as the subsequent fractional increase in the number of W —* pv candidate events.

The standard anti-isolation sample is used to obtain the QCD background event

fraction in W —► ev events, and the uncertainties are combined to give a back­

ground fraction of (1.35 ± 0.28)%.

The QCD rax distribution, shown in figure 7.10, is obtained from the standard

anti-electron sample with the additional requirement of E /p < 3, since QCD

events in the E /p tail have a different mT distribution. The m T distribution is

corrected for electroweak contamination, and fitted with a Gaussian function in

the region 50 < mx < 7 0 GeV and a Landau function in the region 70 < mx <

2 0 0 GeV.

| 102 in(0ca>>in

100 150 200 nV (GeV)

Figure 7.10: The fitted W —» ev multi-jet mT distribution obtained with two of F'had/F'em > 0.07, \Az\ > 0.5 cm and Xstrip > 10 requirements.

171

The uncertainty on Tw from QCD background events in W —» eu candidate

events, shown in table 7.10, is estimated as the change in Tw from varying the

background fraction by its uncertainty and the shape of the m T distribution.

The parameterised shape is varied by each fit parameters’ uncertainty, the elec­

troweak contamination uncertainty and the anti-electron definition. In addition

to the three anti-electron definitions described above, the standard anti-electron

sample with the additional E /p requirement of candidate electrons (E/ p < 1.3)

is used to evaluate the shape uncertainty.

fraction fit parameters EWK anti-electron tota80 11 14 5 25 3185 11 14 5 25 3190 15 16 6 2 2 32

1 0 0 2 2 2 1 1 2 25 411 1 0 26 27 1 0 30 49

Table 7.10: Systematic uncertainties for W —»• ev, in MeV, from QCD background events.

172

Chapter 8

R esults

The value for Tw is extracted from the ttit distribution of W —> eis and W —> /iv

candidate events, described in chapter 4, by varying the Tw input value for the

event generator, described in chapter 5, and minimising the negative binned log

likelihood [63] of the simulated distribution, described in chapter 6 , added to

the background events, described in chapter 7. The simulated ttt-t distribution is

normalised to that of candidate events in the region 50 < raT < GeV and

fitted in the region ra^ < < 200 GeV. The systematic uncertainty associated

with the event simulation and background estimation are determined for the

region < m t < 200 GeV, for ra^ values of 80, 85, 90, 100 and 110 GeV.

The value of used in the measurement of Tw is chosen to minimise the

combined systematic and statistical uncertainty associated with the fit to the

distribution in the region m ^< < 200 GeV. The evaluation of and the

subsequent Tw measurement is described below.

173

8.1 Fit region

The systematic uncertainties associated with the event generation, detector sim­

ulation and background event estimation are summarised in table 8 .1 .

W —> ez/ W —> fiis80 85 90 1 0 0 1 1 0 80 85 90 1 0 0 1 1 0

PDFs 2 1 2 2 2 0 2 1 26 2 1 23 2 0 23 27electroweak corrections 15 14 1 0 6 6 4 6 6 6 6

W boson mass 18 17 9 4 2 18 17 9 4 2

energy-loss simulation 13 13 13 13 13 - - - - -

silicon material scale 3 3 2 1 0 - - - - -electron E t selection 13 13 1 0 8 8 - - - - -lepton u\\ selection 2 2 2 1 0 7 7 6 2 1

lepton rj and selection 3 3 3 3 3 4 4 4 5 5electroweak background 3 3 3 5 8 16 17 14 5 3QCD background 31 31 32 41 49 7 8 1 2 17 2 1

decay-in-flight background - - - - - 18 2 2 27 40 57COT scale - - - - - 29 27 17 8 5COT global resolution - - - - - 29 27 2 1 11 5COT resolution distribution - - - - - 17 17 16 13 1 0

calorimeter scale 29 27 17 7 4 - - - - -calorimeter resolution 46 43 31 11 4 - - - - -calorimeter non-linearity 1 2 13 1 2 1 0 6 - - - - -recoil 60 59 54 38 19 53 51 49 30 14W boson pT 8 8 7 7 6 8 8 7 7 6

total 96 93 79 65 62 80 78 71 62 70

Table 8.1: Summary of all systematic uncertainties, in MeV.

Table 8 .2 shows the total systematic uncertainty, the statistical uncertainty ob­

tained from the negative binned log likelihood fit, and their combined uncer­

tainty. The value of m ^= 90 GeV gives the lowest combined uncertainty for

both W —> eu and W —> (iv events, and is used to obtain the value of Tw-

The total number of background events in W —► e v and W —► f iu and the num­

ber in the fit region 90 < mT < 200 GeV, are shown in table 8.3 with their

distributions shown in figure 8 .1 .

174

W —> ez/ W —> f l V

Tot systematic statistical total systematic statistical total80 96 54 1 1 0 80 60 1 0 0

85 93 54 107 78 61 9990 79 60 99 71 67 981 0 0 65 78 1 0 1 62 90 1091 1 0 62 103 1 2 0 70 116 135

Table 8.2: Combined systematic and statistical uncertainties, in MeV.

fLV evrax region total fit total fitall events 108808 2619 127432 3426W -> r i/ 2 0 2 1 17 2557 2 1

Z 11 6003 216 205 17Z —>• TT 115 2 151 3QCD 291 1 0 1680 99decay-in-flight 145 40

Table 8.3: The number of background events m W —> fiv and W events.

eu candidate

total

— QCD

100 150 200 m,. (GeV)

><3in

— total W > TV

- - - z->wi— QCD— decay-in-flight

50 100 200150m,. (GeV)

Figure 8.1: The distribution of background events for (left) W —> eu and (right) W —► /ii/ events.

8.2 Fit results

The simulated rax distribution is normalised to tha t of W —» eis and W —> fin

candidate events in the region 50 < mT < 90 GeV and fitted in the region

175

90 < mT < 200 GeV. The negative binned log likelihood fit, shown in figure 8 .2 ,

gives

rw = 2118 ± 60stat ± 71sys MeV (8 .1 )

and

Tw = 1948 ± 67stat ± 79sys MeV (8.2)

for W —»■ ev and W —> /xz/ candidate events respectively.

XJ/h d f (fit region) =17/21

X2/nd f (full region) = 21/29<0cUi

100 20015050

X2/n d f ( f i t region) = 19/21

X2/n d f (full region) = 32/29WcIUJ

100 200 itV (GeV)

150

Figure 8 .2 : The m? distribtion of (left) W —► eis and (right) W —► //z/ candidate events fitted to obtain T w

Although separate measurements of Tw are made from W —» eis and W —> [iv

candidate events, the systematic uncertainties associated with the event gener­

ation described in chapter 5 are correlated. Biases in the estimate of the com­

bined value result if the two measurements are combined using the weighted aver­

age. Instead, the results are combined using the Best-Linear-Unbiased-Estimator

(BLUE) method [64], with the correlations of the systematic uncertainties be­

tween the decay channels shown in table 8.4.

176

W —► ev W —> fiis correlationlepton £ t or px scale 2 1 17 12

lepton Et or px resolution 30 26electron energy loss simulation 13recoil model 54 49W boson Pt 7 7 7backgrounds 32 33PDFs 2 0 2 0 2 0

W boson mass 9 9 9electroweak corrections 1 0 6 6

lepton selection 1 0 7total systematic 79 71 27statistical 60 67total combined 99 98 27

Table 8.4: Summary of uncertainties, in MeV, evaluated for = 90 GeV for W —> fiv and W —> en events, together with their correlations between decay chan­nels.

This gives a combined result of

Tw = 2033 ± 73 MeV (8.3)

which is in good agreement with the theoretical prediction of 2093 ± 2 MeV

presented in section 1.3.1, and the world average of 2147 ± 60 MeV [25], with

the measurement presented in this thesis excluded.

8.3 Related measurements

8.3.1 W boson lifetime

Prom equation 1.23, the proper lifetime of the W boson is determined to be

rw = 3.24 ± 0.11 x 10- 25 s (8.4)

177

where the reduced Plank constant h = 6.582 x 10 16 eV-s.

8.3.2 CKM m atrix unitarity

The total W boson decay width is the sum of the partial decay widths to leptons

and quarks, described in section 1.3.1, and can be expressed as

where the sum is over i = u, c and j = d, s, b. This assumes 3 generations of

fermions. The W boson decay width measurement can therefore be used to test

the unitarity of the CKM matrix, described in section 1.1.4, giving

consistent with the unitarity requirement of 2, and the LEP2 value of E |Vij| 2 =

1.993 ± 0.025 obtained from the LEP2 measurement of the branching ratio of

B r(W -> Iv) = (10.84 ± 0.09)% [25].

8.4 Conclusion

A direct measurement of the W boson decay width is obtained with a precision

of 3.6% by fitting the transverse mass distribution of W —► ev and W —► /iv can­

didate events with an integrated luminosity of 350 pb-1 . The measurement of

2033 ± 73 MeV is obtained by combining the separate measurements from the

r w = T(W -> Iv) 3 + 3(1 + SQCD) E i^i2 (8.5)no top

(8 .6)no top

using the values for T(lTr —> Iv) and £QCD presented in section 1.3.1. This is

178

W —* eis and W —► (iv decay channels, accounting for correlated systematic un­

certainties.

This measurement is currently the best single direct W boson decay width mea­

surement. This measurement is in good agreement with the theoretical prediction

of 2093 ± 2 MeV presented in section 1.3.1, and the previous world average of

2147 ± 60 MeV [25]. Combining this measurement with other direct measure­

ments gives a current world average of 2098 ± 48 MeV [21]. In addition, this

measurement is also in agreement with the indirect measurement of 2079 ± 41

MeV [26], described in section 1.3.2.

The measurements of the W boson decay width, both direct and indirect, demon­

strate the consistency of the standard model, although the constraint on the the­

oretical prediction of Tw from the uncertainty on the raw measurement is not

tested at the current precision of Tw measurements.

179

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